# Buoyancy-Tilted Shear Layers and Coherent Structures in Hyperpycnal River Plume Plunging

Haoran Shi, Maria Eletta Negretti, Julien Chauchat, Koen Blanckaert, Ulrich Lemmin, D. Andrew Barry

► To cite this version:

Haoran Shi, Maria Eletta Negretti, Julien Chauchat, Koen Blanckaert, Ulrich Lemmin, et al. Buoyancy-Tilted Shear Layers and Coherent Structures in Hyperpycnal River Plume Plunging. Water Resources Research, 2025, 61 (6), pp.e2024WR039696. 10.1029/2024WR039696. hal-05119068

HAL Id: hal-05119068

https://hal.science/hal-05119068v1

Submitted on 31 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Water Resources Research®

RESEARCH ARTICLE

10.1029/2024WR039696

**Key Points:**

* Shear layers in a hyperpycnal river plume are tilted by buoyancy and form 3D curved interfaces

* Buoyancy-tilted coherent structures develop in shear layers, whose Strouhal number decreases with increasing Richardson number

* Coherent structures are also tilted and curved due to buoyancy and form strongly-mixed "coherent structure regions"

**Correspondence to:**

H. Shi and D. A. Barry,
h.shi@bangor.ac.uk;
andrew.barry@epfl.ch

**Citation:**

Shi, H., Negretti, M. E., Chauchat, J., Blanckaert, K., Lemmin, U., & Barry, D. A. (2025). Buoyancy-tilted shear layers and coherent structures in hyperpycnal river plume plunging. *Water Resources Research*, *61*, e2024WR039696. https://doi.org/10.1029/2024WR039696

Received 12 DEC 2024
Accepted 28 MAY 2025

**Author Contributions:**

**Conceptualization:** K. Blanckaert,
U. Lemmin

**Data curation:** H. Shi

**Formal analysis:** H. Shi

**Funding acquisition:** D. A. Barry

**Investigation:** H. Shi

**Methodology:** H. Shi, M. E. Negretti,
K. Blanckaert

**Resources:** D. A. Barry

**Software:** J. Chauchat

**Supervision:** M. E. Negretti, J. Chauchat,
K. Blanckaert, U. Lemmin, D. A. Barry

**Writing – original draft:** H. Shi

**Writing – review & editing:**

M. E. Negretti, J. Chauchat, K. Blanckaert,
U. Lemmin, D. A. Barry

© 2025. The Author(s).
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.

<figure><img src="image_2.png" /></figure>

Buoyancy-Tilted Shear Layers and Coherent Structures in
Hyperpycnal River Plume Plunging

H. Shi¹,² ID, M. E. Negretti³, J. Chauchat³,⁴ ID, K. Blanckaert⁵, U. Lemmin², and D. A. Barry² ID

¹School of Ocean Sciences, Bangor University, Menai Bridge, Anglesey, UK, ²Ecological Engineering Laboratory (ECOL), Environmental Engineering Institute (IIE), Faculty of Architecture, Civil and Environmental Engineering (ENAC), École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, ³Université Grenoble Alpes, CNRS, Grenoble INP, LEGI UMR 5519, Grenoble, France, ⁴Université Grenoble Alpes, INRAE, CNRS, Grenoble INP, IRD, IGE, Grenoble, France, ⁵Research Unit Hydraulic Engineering and Environmental Hydromechanics, Institute of Hydraulic Engineering and Water Resources Management, Technische Universität Wien, Vienna, Austria

**Abstract** Shear layers and corresponding Kelvin Helmholtz-type Coherent Structures (KHCS) can be generated by rivers discharging into laterally-unconfined quiescent open water bodies (e.g., lakes). When the river discharge has a greater density than the receiving water due to temperature and/or suspended sediment, both the shear layers and KHCS will be influenced by the negative buoyancy of the plume and thus become highly three-dimensional (3D). The present study uses a turbulence-resolving Computational Fluid Dynamics model based on Large Eddy Simulation to simulate the nearshore flow fields of a hyperpycnal river plume entering an unconfined quiescent ambient. Shear layers are observed at both sides of the plume and their growth is suppressed by negative buoyancy arising from the greater density of the river plume. The plume-ambient interface is tilted by the negative buoyancy and is transformed into a curved face. As a result, the shear layer is also tilted and shear-induced vorticity progressively changes its direction from vertical near the water surface to transversal near the bottom. Tilted along with the shear layers, KHCS present unique 3D subsurface structures and create strongly mixed and curved "coherent structure regions" in transects. Quadrant analysis shows that the "Ejection" and "Sweep" events associated with KHCS dominate the local mass and momentum exchange between the plume and ambient water. At the plume-ambient interface, the KHCS generate near-periodic velocity fluctuations whose non-dimensionalized frequency (Strouhal number) decreases with increasing local Richardson number.

1. Introduction

Hyperpycnal river plumes are generated when negatively buoyant river inflows enter a receiving ambient, for example, when a river with relatively low temperature and/or high sediment load flows into a lake (Best et al., 2005; Blanckaert et al., 2024; De Cesare et al., 2001; Piton et al., 2022; Soulignac et al., 2021; Spigel et al., 2005; Vandekerkhove et al., 2020). As described previously (Hogg et al., 2013; Shi et al., 2022; Thorez, Lemmin, et al., 2024), a hyperpycnal river plume converges on the water surface and establishes a triangular pattern when the receiving water body is laterally unconfined (Figure 1a). Such a triangular pattern was also observed in laboratory experiments by Shi et al. (2022), who reproduced the flow pattern of the Rhône River (Lake Geneva) nearfield plume in a laboratory model based on geometric and Froude scaling (Figure 1b). Near the bottom, however, the plume spreads laterally due to its negative buoyancy (Figure 1c). Upon reaching the vertex of the surface triangle, a hyperpycnal river entirely plunges below the water surface and transforms into an underflow (Hogg et al., 2013; Piton et al., 2022; Shi et al., 2022; Thorez, Lemmin, et al., 2024). Field measurements taken in the Rhône River nearfield show that underflow discharge can be more than twice as much as the initial river flow at the river mouth due to ambient water entrainment in the plunging region; this results in a significant decrease in both density and velocity (Blanckaert et al., 2024; Thorez, Blanckaert, et al., 2024). Understanding mixing processes in the plunging region is essential in order to evaluate sediment behavior in the coastal region (Thorez, Lemmin, et al., 2024) and to estimate the characteristics of the subsequent underflow (Blanckaert et al., 2024; Cortés et al., 2014; Rueda et al., 2007). In particular, plunging mixing determines whether the subsequent underflow will transform into an interflow in the pycnocline (Piton et al., 2022; Soulignac et al., 2021) or break through the pycnocline and continue as an underflow in the hypolimnion (Blanckaert et al., 2024).

At the lateral sides of the surface triangle, vertically-oriented turbulent coherent structures are evident in both field observations and laboratory experiments (Figures 1a and 1b). These intermediate scale coherent structures are
<figure><img src="image_3.png" /><figcaption>**Figure 1.** (a) Satellite image of the nearfield of the hyperpycnal Rhône River plume entering Lake Geneva (September 2018). The river mouth is at the bottom of the image. (b) Surface image of the inflow from a channel (at the bottom of the image) entering into a laterally unconfined tank, captured during the laboratory experiment of Shi et al. (2022). The inflow was visualized by injecting Rhodamine and illuminated by a laser system (see their Figure 4a for details). (c) Sketch of a hyperpycnal river plume cross-section. The location of this cross-section is indicated by the yellow dash-dotted line in (b). (d) Sketch of a neutrally-buoyant river plume cross-section. In panels (c, d), the river plume is colored black, and the ambient water is white. The red dashed lines mark the interface between the plume and ambient water which is curved in the case of the hyperpycnal river plume.</figcaption></figure>

generated in a shear layer at the plume-ambient interface that results from the velocity gradient between the river plume and ambient waters. Similar structures have been widely observed and studied in other geophysical flow cases with shear layers, for example, compound channel flows (Proust et al., 2017; Stocchino & Brocchini, 2010), tidal flows (Farmer et al., 2002; Shi et al., 2024; White & Helfrich, 2013), and river confluences (Biron et al., 2019; Lewis & Rhoads, 2015). These shear-induced vertically oriented coherent structures are classified as Kelvin Helmholtz-type Coherent Structures (KHCS) and are known to play a significant role in mass and momentum transfer, including the exchange of sediment, pollutants, and nutrients across shear layers.

Shi et al. (2024) analyzed the case where the river plume and receiving ambient have similar densities. Development of the shear-layer and shear-induced coherent structures was elucidated by both laboratory experiments and numerical modeling. Shear layers and associated coherent structures also develop in hyperpycnal river plumes, although the flow field is more complex because of the negative buoyancy. Since hyperpycnal river plumes slump laterally, the interface between the plume and the ambient is not vertical as would be the case when there is no density difference; instead, it is curved outward (red dashed lines/curves in Figures 1c and 1d). With a curved interface, negative buoyancy is expected to influence the development of the shear layers and shear-induced intermediate scale coherent structures. However, its influence on the characteristics and development of shear layers and coherent structures in such a configuration has yet to be investigated. These coherent structures are known to contribute significantly to mixing between the river plume and ambient waters. Understanding their turbulent characteristics is therefore essential when modeling the transport of river-borne materials such as pollutants and sediments. Buoyancy-influenced shear layers and corresponding coherent structures are also major features of other geophysical flow cases (e.g., river confluences (Duguay et al., 2022), and the separation point of a tidal flow passing an island (Farmer et al., 2002)).
Details of the shear layers and intermediate scale coherent structures are difficult to measure in the field because of the complexity of the hydrodynamic environment of a hyperpycnal river plume (e.g., unsteady river inflow, complex topography, background currents, wind effects) and the limited time-space resolution of measuring equipment. These difficulties can be partially resolved in laboratory experiments, but laboratory measuring techniques can only provide data sets along a line (using Acoustic Doppler Velocity Profilers) or in a plane (using Particle Image Velocimetry), which cannot fully resolve three-dimensional (3D) coherent structures.

This study uses a turbulence-resolving Computational Fluid Dynamics (CFD) model based on Large Eddy Simulation (LES) to simulate the nearshore hydrodynamics of a hyperpycnal river plume entering a laterally-unconfined quiescent open water body. This approach provides continuous high-resolution 3D data sets of the velocity and density fields and thus captures for the first time the development of shear layers and coherent structures. The following questions are addressed:

* What is the influence of negative buoyancy on shear layer development?

* What are the 3D characteristics of shear-induced coherent structures observed on the water surface (Figures 1a and 1b)? How do they differ from the neutrally buoyant case in terms of shape and frequency (i.e., Strouhal number)?

## 2. Numerical Model

Details of the numerical model used in the present study are given in Shi et al. (2022, 2024). The model solves the incompressible Navier-Stokes equations with a transient solver “BuoyantBoussinesqPimpleFoam” provided by the open-source CFD package: OpenFOAM (De Lorenzis & Düster, 2020; Jasak, 2009; Weller et al., 1998). This solver has been extensively validated for buoyancy-driven turbulent currents (e.g., Cintolesi et al., 2019; Habibi et al., 2022; Shi et al., 2022). LES is applied to resolve large and intermediate scale turbulence, while small scales are modeled by a dynamic one-equation subgrid-scale model (Kim & Menon, 1995). This approach balances the need to capture energy-containing turbulent structures with computational feasibility.

The computational domain, which is identical to that of Shi et al. (2022, 2024), consists of a rectangular inlet channel ($B_0 = 2$ m wide, $H_0 = 0.08$ m deep) connected to a receiving tank with an 8° sloping bottom (Figure 2). This numerical domain is the same as the experimental setup of Shi et al. (2022), which is geometrically similar (1:60 scale) to the Rhône River entering Lake Geneva (Figure 1a). An upward looking Cartesian coordinate system with its origin (0, 0, 0) at the water surface and at the downstream end of the inlet channel is applied. Longitudinal ($u = \bar{u} + u'$), transversal ($v = \bar{v} + v'$), and vertical velocities ($w = \bar{w} + w'$) follow the x, y, and z directions, respectively. Time-averaged values are denoted by overbars, and fluctuations by primes. The mesh used in the model has a resolution of 0.007 m × 0.01 m × 0.005 m, which is sufficient to capture intermediate scale coherent structures (Shi et al., 2024) and can resolve ~95% of the total turbulent kinetic energy. Near the bottom boundary, the grids were further refined to reach $\delta_z^+ = u_*\delta_z/\nu \approx 1$ to resolve the viscous sublayer, where $\delta_z$ denotes the vertical size of the cell close to the bottom boundary and $u_*$ is the shear velocity. With this setup, the model was validated with experimental data when the river plume is negatively buoyant (Shi et al., 2022) and when the inflow is neutrally buoyant (Shi et al., 2024). Therefore, this model is suitable to simulate the mean fields of the river plume and the characteristics of intermediate scale coherent structures.

Four numerical cases are investigated where the inflow relative density difference between the river plume and ambient water ($R_0$) increases from 0% to 0.2%, thereby decreasing the inflow densimetric Froude number, $Fr_{d-0}$, from $\infty$ to 2 (Table 1). This range of $Fr_{d-0}$ is representative of natural hyperpycnal river plumes reported in the literature (Hogg et al., 2013; Shi et al., 2022; Spigel et al., 2005). Hereinafter, the subscript “0” denotes the inflow parameters at the river mouth. The inflow relative density difference is defined as $R_0 = (\rho_0 - \rho_a)/\rho_a$, where $\rho_0$ denotes the density of the inflowing river plume, and $\rho_a$ is the density of ambient water. The inflow densimetric Froude number is $Fr_{d-0} = Fr_0/\sqrt{R_0} = U_0/\sqrt{gH_0R_0}$, where $U_0$ and $H_0$ are the inflow bulk velocity and water depth of the river inflow, respectively. Shi et al. (2022) reported that the plunging hydrodynamics of the hyperpycnal river plume are controlled by the inflow densimetric Froude number ($Fr_{d-0}$), not the Froude number ($Fr_0$). Therefore, $Fr_0$ is fixed at 0.08, which falls within the range reported for natural hyperpycnal river plumes, that is, between 0.02 and 0.2 (Hogg et al., 2013; Shi et al., 2022; Spigel et al., 2005; Thorez, Lemmin, et al., 2024). In addition to velocity, the numerical model also outputs the 3D concentration field, which is directly related to fluid density. Concentration ($c$) indicates the proportion of fluid originating from the river plume, that is, $c = 1$ for
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Geometry used in the numerical model, consisting of an inlet channel and a receiving tank with a sloping bottom. Black: river plume; Silver: ambient water. The same geometry was applied as in our previous studies (Shi et al., 2022, 2024).</figcaption></figure>

the inflow and $c = 0$ for the ambient water. Thus, the fluid density follows $\rho \approx c\rho_0 + (1 - c)\rho_a$. The relative density difference is $R = (\rho - \rho_a)/\rho_a \approx cR_0$. We will mainly discuss concentration field, $c$, instead of density field, $R$, since both are equivalent.

# 3. Results and Discussion

## 3.1. Shear Layers and Coherent Structures Observed at the Water Surface

Shear layers can be formed by velocity gradients between two fluids. For the four numerical cases studied here, a velocity gradient between the river plume and ambient water is observed (Figure 3i) in the time-averaged velocity distribution on the water surface ($z/H_0 = 0$, nondimensionalized by $U_0$). The time-averaged concentration field on the water surface shows a similar pattern as the velocity field (Figure 3ii). The distribution of turbulent kinetic energy $k = 0.5(u'u' + v'v' + w'w')$ on the water surface (nondimensionalized by $U_0^2$) highlights the shear

<caption>**Table 1**</caption>

Summary of Parameters Used in Numerical Cases 1–4

<table><thead><tr><th>Case</th><th>Q<sub>0</sub> (L s<sup>-1</sup>)</th><th>U<sub>0</sub> (m s<sup>-1</sup>)</th><th>R<sub>0</sub></th><th>Fr<sub>0</sub></th><th>Fr<sub>d-0</sub></th><th>β (°)</th><th>x<sub>m</sub>/B<sub>0</sub></th><th>M at edge (%)</th></tr></thead><tbody><tr><td>1</td><td>12</td><td>0.075</td><td>0</td><td>0.08</td><td>∞</td><td>8</td><td>1.24</td><td>200</td></tr><tr><td>2</td><td>12</td><td>0.075</td><td>1.12 × 10<sup>-4</sup></td><td>0.08</td><td>8</td><td>8</td><td>1.07</td><td>54</td></tr><tr><td>3</td><td>12</td><td>0.075</td><td>4.5 × 10<sup>-4</sup></td><td>0.08</td><td>4</td><td>8</td><td>0.72</td><td>38</td></tr><tr><td>4</td><td>12</td><td>0.075</td><td>2 × 10<sup>-3</sup></td><td>0.08</td><td>2</td><td>8</td><td>0.45</td><td>49</td></tr></tbody></table>

*Note.* *Q*: discharge; *U*: bulk velocity; *B*: plume width; *R*: relative density difference; *Fr*: Froude number; *Fr<sub>d</sub>*: densimetric Froude number; *β*: slope of the bed; *x<sub>m</sub>*: longitudinal coordinate where the shear layers at two sides meet at the centerline. The subscript “0” denotes the inflow parameters at the river mouth. “M at edge” denotes the ratio of the mixing term at the edge of the coherent structure region compared with the surface (Section 3.3.2).
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Flow fields near the water surface (time-averaged over 100 s). Columns (a–d) present Cases 1–4, respectively. Panels (i): Longitudinal velocity. Panels (ii): Concentration field. Panels (iii): Turbulent kinetic energy. Dash-dotted lines mark the interface between the river plume and ambient water. Black dashed line in Panel (iii) is perpendicular to the plume-ambient interface whose intersection with the interface is located at $x = 0.5x_m$. Panels (iv): Concentration profile along the surface centerline ($y = 0, z = 0$). Crosses: location where shear layers from both sides meet at the centerline. Legends gives the parameter ranges.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** (a) Typical velocity profile across the plume-ambient interface (along the black dashed lines in Figure 3iii). (b) Variation of the momentum thickness on the water surface with the development of the shear layer. Cases 1–4 are color coded (see legend) and are labeled as (Case number, $Fr_{d-0}$) in the legends.</figcaption></figure>

layers (Figure 3iii). The time-averaged concentration contour $\bar{c} = 0.5$ is used to represent the interface between the river plume and ambient water; it is located at the center of the shear layers (Figure 3iii). In the nearfield of the plume, shear layers converge toward the inflow centerline. In Case 1, this convergence of the river plume is induced by the sloping bottom boundary (Shi et al., 2024). As a result of the lateral slumping of the hyperpycnal river plume in Cases 2–4, shear layer convergence on the surface is even stronger in these cases (Hogg et al., 2013; Shi et al., 2022). Following Shi et al. (2024), we define a curvilinear coordinate system ($x_n, y_n$) along the plume-ambient interface (Figure 3iii), with corresponding velocities $u_n = \bar{u}_n + u'_n$ and $v_n = \bar{v}_n + v'_n$, which are, respectively, tangent and perpendicular to the interface curve. At a certain distance from the inlet ($x = x_m$), the two lateral shear layers meet at the centerline, which is estimated as the location where centerline concentration starts to decrease (Figure 3iv).

Since the system is symmetrical on both sides of the centerline ($y = 0$), only one side ($y < 0$) is considered in the following analysis. Along a representative line perpendicular to the interface curve (dashed line in Figure 3iii, which intersects with the interface curve at $x = 0.5x_m$), the velocity increases from $\bar{u}_{n-a}$ (at $y_{n-a}$) inside the ambient water to $\bar{u}_{n-c}$ (at $y_{n-c}$) inside the river plume (Figure 4a). Based on the velocity profile, we calculate the momentum thickness of the shear layer as White and Nepf (2007):
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Instantaneous fields near the water surface. Columns (a–d) present Cases 1–4, respectively. Panels (i): Concentration fields. Panels (ii): Vorticity fields. Panels show surface patterns. Legends give the parameter ranges.</figcaption></figure>

$$ \theta_s = \int_{y_{n-a}}^{y_{n-c}} \left[ \frac{1}{4} - \left( \frac{\bar{u}_n - \bar{u}_{n-a}}{\bar{u}_{n-c} - \bar{u}_{n-a}} \right)^2 \right] dy_n \quad (1) $$

where $\bar{u}_{n-a}$ is the arithmetic mean of $\bar{u}_{n-c}$ and $\bar{u}_{n-c}$. The variation of the shear layer momentum thickness along the interface curve (nondimensionalized by $B_0$) is depicted in Figure 4b. With increasing distance from the river mouth, the momentum thickness increases as shear layers develop due to mixing. However, its growth is suppressed when negative buoyancy is introduced.

Vertically-oriented coherent structures are observed on both sides of the surface triangle (Figure 5). These structures inside the shear layers are induced by the velocity gradient between the river plume and ambient water, and thus are supposed to be KHCS (Brown & Roshko, 1974). They are smaller in cases with smaller inflow densimetric Froude number because the growth of the shear layers is suppressed (Figure 4). The rotation of these structures shown by the vorticity field is clockwise ($\omega_z < 0$) in the right-side shear layer ($y < 0$) and counterclockwise ($\omega_z > 0$) in the left side shear layer ($y > 0$) (Figure 5ii). The sense of rotation is determined by the

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Columns (a–d) present Cases 1–4, respectively. Panels (i): Time-averaged vertical vorticity. Panels (ii): Vorticity stretching in the vertical direction induced by mean velocity field. Panels (iii): Magnitude of vertical vorticity tilted into horizontal directions by the mean velocity field. Panels show surface patterns. Legends give the parameter ranges.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Columns (a–d) present Cases 1–4, respectively. Panels (i): time-averaged (over 100 s) iso-surface $\bar{c} = 0.5$, colored by the mixing term $\overline{c'c'}$; legend gives the color code. Panels (ii): transient iso-surface $c = 0.5$, colored by transient transversal velocity at the same time; legend gives the color code. The sloping bottom boundary is indicated by dashed lines. Red parallelograms in (i): planes perpendicular to the plume-ambient interface curve on the water surface with an intersection at $x = 0.5x_m$. KHCS: Kelvin Helmholtz-type Coherent Structures. $x_m$: longitudinal position where the two shear layers meet at the centerline.</figcaption></figure>

velocity gradient and indicated by the time-averaged vertical vorticity (Figure 6i). Vorticity is controlled by a balance between advection, baroclinic production, dissipation, stretching, and tilting (Cantero et al., 2007; Yuan & Horner-Devine, 2017). Stretching and tilting, which are absent in two-dimensional flows (Tennekes & Lumley, 1972) result in lengthening and tilting of vorticity tubes (Farmer et al., 2002), and thus reflect the three-dimensionality of the flow. Figures 6ii and 6iii presents the vorticity stretching and tilting induced by the mean velocity field:

$$sh = \bar{\omega}_z \frac{\partial \bar{w}}{\partial z} \qquad (2a)$$

$$|tl| = \sqrt{\left(\bar{\omega}_z \frac{\partial \bar{u}}{\partial z}\right)^2 + \left(\bar{\omega}_z \frac{\partial \bar{v}}{\partial z}\right)^2} \qquad (2b)$$

Here, we consider only the vertical stretching of vorticity (sh) and the magnitude of vertical vorticity tilted in the horizontal directions (tl). In Case 1 (no buoyancy contrast), sh can hardly be seen, and mainly represents noise arising from the heterogeneity of the flow fields (Yuan & Horner-Devine, 2017). Vorticity stretching that enhances vertical vorticity, that is, sh > 0 when y > 0 and sh < 0 when y < 0, is observed in the negatively buoyant cases. This indicates that the vorticity tube is stretched. Similarly, |tl| is weak in Case 1, but becomes more evident in cases with larger density contrast, indicating the tilting of the vortex tubes.
<figure><img src="image_10.png" /><figcaption>**Figure 8.** 3D sketch of the development of vortex tubes in (a) neutrally buoyant and (b) negatively buoyant cases. Vortex tubes stretch with distance from the river mouth and tilt in the negatively buoyant case. The brown region bounded by the dash-dotted curve represents the river plume on the water surface. The gray plane represents the slopping bottom boundary. Red-blue arrow pairs indicate the sense of rotation of the vortices.</figcaption></figure>

## 3.2. 3D Shape of the Shear Layer and Coherent Structures

Figure 7i shows the time-averaged iso-surface $\bar{c} = 0.5$ in the four numerical cases, representing different evolutions of the plume-ambient interface and the underwater shear layer. The 3D plume-ambient interface ($\bar{c} = 0.5$) is vertically oriented in Case 1. However, it is curved in the remaining cases because negative buoyancy induces lateral slumping of the river plume. Comparing the four numerical cases, the curvature of the 3D plume-ambient interface increases as the inflow densimetric Froude number decreases. When the plume-ambient interface tilts or stretches, the accompanying turbulent structures also tilt or stretch (Farmer et al., 2002). The subsurface 3D shape of the coherent structures observed on the water surface is represented by the transient iso-surface $c = 0.5$ (Figure 7ii), which is rough and wrinkled. Each wrinkle on the water surface indicates a coherent structure that extends to a certain depth below the surface and establishes a local pattern of alternating positive-negative transverse velocity pairs (Figure 7ii). The iso-surface contour ($\bar{c} = 0.5$) shown in Figure 7i is colored by the mixing term $(c'c')$ and thus quantifies the strength of local mixing (Duguay et al., 2022). In Cases 2–4, the mixing term is larger in the region where coherent structures are found, indicating stronger local mixing compared with other parts of the plume-ambient interface. Regions with relatively high or low local mixing strengths are highlighted by brown (high, close to water surface) and green (low, close to bottom) arrows in Figures 7b–7d. Such an observation is not clear in Case 1 (neutral buoyancy).

The tilting mechanism induced by negative buoyancy is illustrated in a 3D sketch (Figure 8), which compares the vortex tube development in neutrally and negatively buoyant cases. Without the density contrast, the plume-ambient interface is vertical, and the vortex tube is lengthened only due to the offshore increase of water depth (Figure 8a). In the negatively buoyant cases (Figure 8b), the vortex tube is tilted and lengthened because the plume-ambient interface is tilted and stretched due to buoyancy. This explains the larger stretching and tilting terms on the water surface in these cases, as described previously in Section 3.1. Moreover, as the vortex tube is lengthened, vortex stretching suppresses the offshore growth of the shear layer, which explains the smaller momentum thickness observed on the water surface in larger density contrast cases (Figure 4b).

## 3.3. Characteristics of the Coherent Structures in Transects

A representative transect is selected to analyze the characteristics of coherent structures (red parallelograms in Figure 7i). This transect is perpendicular to the plume-ambient interface curve on the water surface (solid curve in Figure 8) with an intersection at $x = 0.5x_m$ (referred to as PT$_5$ hereinafter). Flow patterns in PT$_5$ for all flow cases are shown in Figure 9. The velocity difference between the plume and ambient waters indicates strong shear between the two fluids (Figure 9i). Inside PT$_5$, the plume-ambient interface is determined based on the concentration contour $\bar{c} = 0.5$ (after smoothing, dash-dotted lines in Figure 9), consistent with the velocity distribution (Figure 9i).

### 3.3.1. Coherent Structure Region

Under the influence of coherent structures, concentration signals are expected to be correlated in the region where the structures exist. We define a “coherent structure region” based on the Pearson correlation function, $r_{cc}$ (Nezu & Nakagawa, 1993; Shi et al., 2024). This analysis is carried out in PT$_5$. The time-series of the concentration fluctuations $c'$ (100 s at 10 Hz) at a point located on the plume-ambient interface curve and close to the water
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Subplots (a–d) present Cases 1–4, respectively. Data for selected representative transects that are perpendicular to the plume-ambient interface curve on the water surface at intersection $x = 0.5x_m$ (PTs; red parallelograms in Figure 7i). Panels (i): Time-averaged velocity distribution. At the blue circles located on the plume-ambient interface and near the water surface, the reference signal for the Pearson correlation function is taken; for details, see Section 3.3.1. Panels (ii): Distribution of the correlation function (Equation 3). The red dashed curve denotes the contour $r_{cc} = 0.3$, representing the coherent structure region. The red circles located at $y_p = 0$ mark the central point of the coherent structure region. The $y_p$ coordinate is perpendicular to the plume-ambient interface and crosses the central point. This coordinate is identical to $y_n$ in Case 1 and forms an angle $\gamma$ with $y_n$ in the other cases. Panels (iii): Profiles of time-averaged density (red line) and velocity (blue line) along the $y_p$ coordinate, which were used to calculate Richardson numbers (Ri). Panels (iv): Distribution of the mixing term $(c'c')$. The dotted lines in (i) and (iv) mark the contours $\bar{c} = 0.05$ and $\bar{c} = 0.95$. The black dash-dotted curves in (i), (ii), and (iv) show the plume-ambient interface $\bar{c} = 0.5$.</figcaption></figure>

surface is selected as the reference signal (at $y_n \approx 0$ and $z \approx 0$; blue circles in Figure 9i). The Pearson correlation function between this reference signal and time-series of $c'$ at other locations inside the selected transects (over the same period) is given by:
<figure><img src="image_12.png" /><figcaption>**Figure 10.** (a) Variations of the mixing term along the plume-ambient interface in PT₅ for Cases 1 and 4. Solid part of each curve is located inside the coherent structure region and dashed part is located outside the region. Legend gives (Case number, Frd-0). (b, c) Variations of the shear-induced vorticity direction (blue arrows) along the plume-ambient interface in PT₅ (red dash-dotted line/curve) for (b) Case 1 and (c) Case 4. Black dashed line: sloping bottom boundary. s_c represents the distance from the surface point (blue circles) along the interfacial line/curve.</figcaption></figure>

$$r_{cc}(y_n, z) = \frac{\overline{c'(y_n, z, t)c'(y_n \approx 0, z \approx 0, t)}}{\sqrt{[c'(y_n, z, t)]^2 [c'(y_n \approx 0, z \approx 0, t)]^2}} \quad (3)$$

The value of $r_{cc}$ (Figure 9ii) decreases with increasing distance from the reference point (blue circle in Figure 9i). A threshold of $r_{cc} = 0.3$ (Shi et al., 2024) is used to determine coherent structure regions as illustrated by the red dashed curves in Figure 9ii. With increasing negative buoyancy from Case 1 to Case 4, the distortion of the coherent structure region increases. This is due to the tilting of the plume-ambient interface and the vortex tube.

A portion of the plume-ambient interface curve in PT₅ (black dash-dotted lines, Figure 9) is located inside the coherent structure region (Figure 9ii). The geometric midpoint of this portion is defined as the central point of the corresponding coherent structure region (red circles in Figure 9ii). We define a new coordinate ($y_p$) to represent the direction perpendicular to the dash-dotted plume-ambient interface curve, which intersects it at the central point. This coordinate forms an angle $\gamma$ with $y_n$, which is zero in Case 1 but is larger in the other cases. Based on concentration and velocity profiles along the $y_p$ coordinate (Figure 9iii), the local Richardson number on the central point is calculated as:

$$Ri = R_0 \sin \gamma \frac{\partial \bar{c}/\partial y_p}{\partial \bar{u}_n/\partial y_p} \quad (4)$$

All calculated Richardson numbers are ≤0.25, thus indicating that the plume-ambient interface is unstable and likely subject to Kelvin Helmholtz instability (e.g., Negretti et al., 2017; Parker et al., 2019; Yuan & Horner-Devine, 2017).

### 3.3.2. Local Mixing Strength

The distributions of the mixing term $\overline{c'c'}$ in PT₅ are illustrated in Figure 9iv. It is confirmed that in Cases 2–4 the mixing term is larger in the coherent structure region (red dashed curves) than in other areas. We analyzed the variations of the mixing term with increasing distance from the surface point for Cases 1 and 4 (Figure 10a). This distance was calculated along the plume-ambient interface in PT₅ (Figures 10b and 10c: red dash-dotted curve) and denoted as $s_c$ (nondimensionalized by $H_0$ in Figure 10a). The mixing term is nondimensionalized by its magnitude at the starting point of the interfacial curve (i.e., $z = 0$), and denoted as $M$. Each curve is divided into two portions: inside (solid) and outside (dashed) the coherent structure region. A decreasing mixing term is observed in negatively buoyant Case 4. The value of the mixing term at the edge of the coherent structure region is only 49% of the value at the surface. Such a decrease of local mixing is also observed in Cases 2 and 3 (Table 1, 40%–50%) but does not exist in Case 1 without buoyancy. The decreasing trend of the mixing term in negatively buoyant cases is explained by the tilting of the shear layer and the plume-ambient interface. Figures 10b and 10c compares the plume-ambient interfaces and time-averaged vorticity direction along them for Cases 1 and 4. The vorticity direction is determined from the ratio between its vertical ($\partial \bar{u}_n/\partial y_n$) and transversal ($\partial \bar{u}_n/\partial z$) components. The direction of the vorticity approximately follows the -z direction in Case 1 with neutral buoyancy. In contrast, this direction changes gradually from -z to -y_n along the plume-ambient interface in Case 4. This means that vorticity develops a significant horizontal component, allowing the stratification between the plume and the ambient water to act on the shear layer. This also suggests that negative buoyancy suppresses local mixing and the development of shear layers and shear-induced coherent structures.
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Columns (a–d) present Cases 1–4. Panels (i) and (ii): Time series of concentration *c* and velocity *v*<sub>*p*</sub>, respectively, along the representative line perpendicular to the plume-ambient interface (*y*<sub>*p*</sub> in Figure 9ii). Panels (iii) and (iv): Time-averaged covariance terms *u*'<sub>*n*</sub>*v*'<sub>*p*</sub> and *c*'<sub>*v*<sub>*p*</sub></sub> along the representative line in different quadrant events: Q1 (red), Q2 (orange), Q3 (blue), and Q4 (green).</figcaption></figure>

3.3.3. Quadrant Analysis

Quadrant analysis (Bagherimiyab & Lemmin, 2018; Nezu & Nakagawa, 1993) was also performed to detect the shear-induced coherent structures at PT₅. Figures 11i and 11ii shows the concentration (c') and velocity (v'_p) fluctuations along the y_p coordinate.

Two covariance terms (u_n' v_p' and c_p' v_p') were proposed in the literature to describe momentum and mass exchange in each quadrant (e.g., Variano & Cowen, 2013; Wallace, 2016; White & Nepf, 2007). Following White and Nepf (2007), the momentum exchange term is nondimensionalized by shear velocity u_s^2 = -u_n' v_p' at y_p = 0. Similarly, the mass exchange term c_p' v_p' is nondimensionalized by t_*^2 = -c_p' v_p' at y_p = 0. As shown in Figures 11iii and 11iv, ejection (Q2: u'_n < 0 and v'_p > 0) and sweep (Q4: u'_n > 0 and v'_p < 0) events are much stronger than interaction events (Q1: u'_n > 0 and v'_p > 0; Q3: u'_n < 0 and v'_p < 0). This is a common feature of KHCS, and also indicates the dominance of coherent structures in momentum and mass exchange across the plume-ambient interface (Shi et al., 2023, 2024; White & Nepf, 2008). In addition, Figure 11 illustrates that the size of the shear layer and the coherent structures is smaller in cases with smaller Fr<sub>d-0</sub>, further highlighting the suppressive effect of negative buoyancy on the growth of the shear layer and coherent structures.

3.3.4. Spectral Analysis

Coherent structures induce periodic velocity fluctuations in shear layers; their period can be detected by power spectral analysis (e.g., Akula et al., 2017; Shi et al., 2023, 2024; Truong & Uijttewaal, 2019). Figure 12 presents the power spectra of the crossing-interface velocity signal v<sub>p</sub> (along the y<sub>p</sub> direction) at the central point of the coherent structure region in PT<sub>5</sub>. The frequency is nondimensionalized into the Strouhal number: St = fθ/ū<sub>n-o</sub> (Ghisalberti & Nepf, 2005; Ho & Huerre, 1984), where the momentum thickness θ and arithmetic mean velocity ū<sub>n-o</sub> are calculated based on the local velocity profile along the y<sub>p</sub> coordinate (Figure 9iii). The Strouhal number corresponding to small-scale eddies that are five times the size of the numerical grid is represented by green lines
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Power spectral density calculated from the crossing-interface velocity signal $v_p$ at the central point of the coherent structure region in $PT_5$ for Cases 1–4 (a–d). The locations of the central points are shown by the red circles in Figure 9ii. Red and blue dashed lines: slopes of the energy cascade –5/3 (blue) and –3 (red). The green vertical line represents the frequency corresponding to eddies that are five times the size of the numerical grid.</figcaption></figure>

in Figure 12; above them, the spectrum might be influenced by the subgrid scale model. All spectra are unimodal with a fundamental peak ($St_{cs}$), which is the dominant frequency of the coherent structures (e.g., Deo et al., 2008; Shi et al., 2023, 2024; White & Nepf, 2008). $St_{cs}$ in $PT_5$ ranges from 0.4 to ~0.8 for the four cases. This means that
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Cases 1–4 (top to bottom). (a) Evolution in the transect shape of the coherent structure region. Selected transects are vertical and are perpendicular to the surface plume-ambient interface curve at intersections ($x/x_m = 0.2, 0.4, 0.6$, and 0.8 corresponding to $PT_2$ to $PT_8$; left to right). The bottom boundary is marked in gray. (b) Increase in size of the coherent structure region transect shape.</figcaption></figure>

the typical periods of the coherent structures are 12 to ~17 s; thus, 6 to ~8 coherent structures can be captured during the 100 s sampling period. Instead of the typical −5/3 slope of an isotropic turbulence energy cascade, the spectra in Figure 12 approximately follow the −3 slope in a subrange after the fundamental peak (solid curve in Figure 12). The slope of this subrange is between −2.9 and −2.6, based on linear regression. This slope is consistent with the energy cascade behavior reported in the literature for shear-induced KHCS (Shi et al., 2023; Truong & Uijttewaal, 2019; Uijttewaal & Booj, 2000), further confirming that these coherent structures are Kelvin Helmholtz-type.

3.4. Development of Coherent Structures

As the distance from the river mouth increases, the shear layer and coherent structures grow in size due to mixing. In addition to $PT_5$, we also detected coherent structure regions in other transects which are perpendicular to the surface plume-ambient interface but with intersections of $x = (0.2-0.8)x_m$. These transects are denoted as $PT_2$ to $PT_8$. The evolution of the coherent structure region transect shape in the different cases (Figure 13a) shows that in Cases 2–4, the transect shape is gradually stretched and tilted, whereas in Case 1, it is only gradually stretched due
<figure><img src="image_16.png" /><figcaption>**Figure 14.** (a) Variations of the Richardson number at the central points of coherent structure regions in transects. The dash-dotted line indicates the critical Richardson number *Ri* = 0.25 for Kelvin Helmholtz-type Coherent Structures (KHCS). (b) Variations of the Strouhal number associated with KHCS as function of the local Richardson number. The dashed line gives the empirical relationship established in this study (for details, see text), with a squared correlation coefficient of 0.8. Cases 1–4 are identified by color/ symbol and are labeled as (Case number, *Fr*<sub>*d*−0</sub>) in legends.</figcaption></figure>

to the vertical extension of the plume (Shi et al., 2024). The size of the transect shape (*A*<sub>c</sub>, nondimensionalized by the inflow cross-sectional area) increases in all cases with increasing offshore distance (Figure 13b), thus indicating the development of the shear layer and coherent structures.

Local Richardson and Strouhal numbers for the central points of the coherent structure regions in transects PT<sub>2</sub> to PT<sub>8</sub> are presented in Figure 14. In Case 1 (with neutral buoyancy), the Richardson number remains zero. In the negative buoyancy cases, however, positive Richardson numbers are observed. Most Richardson numbers are smaller than 0.25 (Figure 14a). The few exceptions in Case 4 (with a maximum value of 0.27) may be due to the uncertainties in computing gradients of velocity and density. In shear layers without any buoyancy effect, a constant Strouhal number close to 0.079 was associated with KHCS (Ho & Huang, 1982; Ho & Huerre, 1984; Shi et al., 2024). In the present study, however, we observe that with increasing local Richardson number, *St*<sub>cs</sub> decreases (Figure 14b), since negative buoyancy limits the extension of the shear layer (i.e., momentum thickness) and coherent structures. This is caused by the two abovementioned mechanisms, that is, (a) vortex stretching suppresses the radius increase of vortex tubes due to mixing (Section 3.2), and (b) negative buoyancy acts on the horizontal component of vorticity and limits the growth of the shear layer (Section 3.3.2). An empirical relationship between *Ri* and *St*<sub>cs</sub> (*St*<sub>cs</sub> = 0.9*Ri*<sup>2</sup> - 0.4*Ri* + 0.079) was obtained based on least square fitting (dashed line in Figure 14b), with mean absolute percentage errors of 10%, 17%, 14%, and 20% for Cases 1–4, respectively.

## 4. Summary and Conclusions

A turbulence-resolving numerical LES model was used to simulate the near field dynamics of a negatively buoyant (hyperpycnal) river plume entering a quiescent unconfined receiving water body. The objective was to characterize and quantify the influence of negative buoyancy on the development of shear layers and shear-induced coherent structures at the interface between the river plume and the ambient water body. Such information is presently lacking, because most previous studies were conducted in laterally confined settings where this interface does not exist. Investigating this interface in detail is crucial for understanding the dynamics of mixing between river and ambient waters.

The high-resolution numerical simulation data show that:

* Negative buoyancy suppresses the growth of the shear layers but enhances the three-dimensionality of the flow field by transforming the plume-ambient interface from a vertical interface to a curved interface.

* Unlike the neutrally buoyant case where shear-induced vorticity is always vertical, the shear-induced vorticity direction changes from vertical near the water surface to transversal near the bottom in negatively buoyant plumes.

* Coherent structures, whose subsurface characteristics are captured for the first time by the numerical simulations, are generated by shear between the plume and the ambient water; these can be identified as KHCS.

* KHCS are found to extend downwards from the water surface with increasing distance from the river mouth. Since the shear layers are tilted, KHCS are distorted in negatively buoyant cases, creating curved coherent structure regions in transects, a feature not observed in the neutrally buoyant case.

* In negatively buoyant cases, local mixing strength inside the coherent structure region is larger than in other areas, indicating the significant effect that KHCS can have on mixing. In transects, the mixing term decreases along the plume-ambient interfacial curve with increasing distance from the water surface. For example, the local mixing term decreases to 40%–50% at the edge of the coherent structure region compared to the surface in the transect PT<sub>5</sub>.

* A quadrant analysis in PT<sub>5</sub> shows that Ejection (Q2) and Sweep (Q4) events induced by KHCS dominate momentum and mass exchange inside shear layers, consistent with previous studies on KHCS. However, we report for the first time that the Strouhal number (nondimensionalized frequency of KHCS) decreases with
increasing local Richardson number due to the influence of negative buoyancy on the shear layer and KHCS development.

These findings on the dynamics of hyperpycnal river plumes are also applicable to other geophysical flow cases where buoyancy-tilted shear layers and KHCS are generated, such as in lakes and oceans. The present results contribute to understanding certain previously unknown mixing processes between two fluids with different velocities and densities, which control the exchange of materials between them. This includes nutrients, pollution and sediments that can significantly affect aquatic eco-environments and subaqueous geomorphologies. Although this study does not consider suspended sediments (whose deposition and resuspension can sometimes modify density contrast and buoyancy), the findings on shear layers, coherent structures and turbulent kinetic energy reported here can guide future research on sediment transport under similar flow conditions.

## Data Availability Statement

Data reported in this paper can be downloaded from the open-access repository (Shi, 2025): https://zenodo.org/records/15276522.

## Acknowledgments

The authors sincerely thank the anonymous reviewers for their constructive comments and suggestions.

## References

Akula, B., Suchandra, P., Mikhaeil, M., & Ranjan, D. (2017). Dynamics of unstably stratified free shear flows: An experimental investigation of coupled Kelvin-Helmholtz and Rayleigh-Taylor instability. *Journal of Fluid Mechanics*, **816**, 619–660. https://doi.org/10.1017/jfm.2017.95

Bagherimiyab, F., & Lemmin, U. (2018). Large-scale coherent flow structures in rough-bed open-channel flow observed in fluctuations of three-dimensional velocity, skin friction and bed pressure. *Journal of Hydraulic Research*, **56**(6), 806–824. https://doi.org/10.1080/00221686.2017.1416686

Best, J. L., Kostaschuk, R. A., Peakall, J., Villard, P. V., & Franklin, M. (2005). Whole flow field dynamics and velocity pulsing within natural sediment-laden underflows. *Geology*, **33**(10), 765. https://doi.org/10.1130/G21516.1

Biron, P. M., Buffin-Bélanger, T., & Martel, N. (2019). Three-dimensional turbulent structures at a medium-sized confluence with and without an ice cover. *Earth Surface Processes and Landforms*, **44**(15), 3042–3056. https://doi.org/10.1002/esp.4718

Blancakaert, K., Vinnà, L. R., Bouffard, D., Lemmin, U., & Barry, D. A. (2024). Field observations reveal how plunging mixing and sediment resuspension affect the pathway of a dense river inflow into a deep stratified lake. *Water Resources Research*, **60**(4), e2023WR036813. https://doi.org/10.1029/2023WR036813

Brown, G. L., & Roshko, A. (1974). On density effects and large structure in turbulent mixing layers. *Journal of Fluid Mechanics*, **64**(4), 775–816. https://doi.org/10.1017/S002211207400190X

Cantero, M. I., Balachandar, S., & Garcia, M. H. (2007). High-resolution simulations of cylindrical density currents. *Journal of Fluid Mechanics*, **590**, 437–469. https://doi.org/10.1017/S0022112007008166

Cintolesi, C., Petronio, A., & Armenio, V. (2019). Turbulent structures of buoyant jet in cross-flow studied through large-eddy simulation. *Environmental Fluid Mechanics*, **19**(2), 401–433. https://doi.org/10.1007/s10652-018-9629-1

Cortés, A., Fleenor, W. E., Wells, M. G., de Vicente, I., & Rueda, F. J. (2014). Pathways of river water to the surface layers of stratified reservoirs. *Limnology & Oceanography*, **59**(1), 233–250. https://doi.org/10.4319/lo.2014.59.1.0233

De Cesare, G., Schleiss, A., & Hermann, F. (2001). Impact of turbidity currents on reservoir sedimentation. *Journal of Hydraulic Engineering*, **127**(1), 6–16. https://doi.org/10.1061/(ASCE)0733-9429(2001)127:1(6)

De Lorenzis, L., & Düster, A. (2020). Modeling in engineering using innovative numerical methods for solids and fluids. *CISM International Centre for Mechanical Sciences* (Vol. 599). Springer International Publishing. https://doi.org/10.1007/978-3-030-37518-8

Deo, R. C., Mi, J., & Nathan, G. J. (2008). The influence of Reynolds number on a plane jet. *Physics of Fluids*, **20**(7), 075108. https://doi.org/10.1063/1.2959171

Duguay, J., Biron, P. M., & Lacey, J. (2022). Impact of density gradients on the secondary flow structure of a river confluence. *Water Resources Research*, **58**(10), e2022WR032720. https://doi.org/10.1029/2022WR032720

Farmer, D., Pawlowicz, R., & Jiang, R. (2002). Tilting separation flows: A mechanism for intense vertical mixing in the coastal ocean. *Dynamics of Atmospheres and Oceans*, **36**(1–3), 43–58. https://doi.org/10.1016/S0377-0265(02)00024-6

Ghisalberti, M., & Nepf, H. (2005). Mass transport in vegetated shear flows. *Environmental Fluid Mechanics*, **5**(6), 527–551. https://doi.org/10.1007/s10652-005-0419-1

Habibi, S., Azadi, A., & Firoozabadi, B. (2022). Large eddy simulation of inclined negatively buoyant jets with sloped beds. *Journal of Hydraulic Engineering*, **148**(11), 04022023. https://doi.org/10.1061/(ASCE)HY.1943-7900.0002017

Ho, C. M., & Huang, L. S. (1982). Subharmonics and vortex merging in mixing layers. *Journal of Fluid Mechanics*, **119**, 443–473. https://doi.org/10.1017/S0022112082001438

Ho, C. M., & Huerre, P. (1984). Perturbed free shear layers. *Annual Review of Fluid Mechanics*, **16**(1), 365–422. https://doi.org/10.1146/annurev.fl.16.010184.002053

Hogg, C. A. R., Marti, C. L., Huppert, H. E., & Imberger, J. (2013). Mixing of an interflow into the ambient water of Lake Iseo. *Limnology & Oceanography*, **58**(2), 579–592. https://doi.org/10.4319/lo.2013.58.2.0579

Jasak, H. (2009). OpenFOAM: Open source CFD in research and industry. *International Journal of Naval Architecture and Ocean Engineering*, **1**(2), 89–94. https://doi.org/10.2478/IJNAOE-2013-0011

Kim, W. W., & Menon, S. (1995). A new dynamic one-equation subgrid-scale model for large eddy simulations. In *33rd Aerospace Sciences Meeting and Exhibit. Reno, NV, U.S.A.* https://doi.org/10.2514/6.1995-356

Lewis, Q. W., & Rhoads, B. L. (2015). Resolving two-dimensional flow structure in rivers using large-scale particle image velocimetry: An example from a stream confluence. *Water Resources Research*, **51**(10), 7977–7994. https://doi.org/10.1002/2015WR017783

Negretti, M. E., Flör, J. B., & Hopfinger, E. J. (2017). Development of gravity currents on rapidly changing slopes. *Journal of Fluid Mechanics*, **833**, 70–97. https://doi.org/10.1017/jfm.2017.696
Nezu, I., & Nakagawa, H. (1993). *Turbulence in open-channel flows*. IAHR, A.A. Balkema. https://doi.org/10.1201/9780203734902

Parker, J. P., Caulfield, C. P., & Kerswell, R. R. (2019). Kelvin–Helmholtz billows above Richardson number 1/4. *Journal of Fluid Mechanics*, 879, R1. https://doi.org/10.1017/jfm.2019.725

Piton, V., Soulignac, F., Lemmin, U., Graf, B., Wynn, H. K., Blanckaert, K., & Barry, D. A. (2022). Tracing unconfined nearfield spreading of a river plume interflow in a large lake (Lake Geneva): Hydrodynamics, suspended particulate matter, and associated fluxes. *Frontiers in Water*, 4, 943242. https://doi.org/10.3389/frwa.2022.943242

Proust, S., Fernandes, J. N., Leal, J. B., Rivière, N., & Peltier, Y. (2017). Mixing layer and coherent structures in compound channel flows: Effects of transverse flow, velocity ratio, and vertical confinement. *Water Resources Research*, 53(4), 3387–3406. https://doi.org/10.1002/2016WR019873

Rueda, F. J., Fleenor, W. E., & de Vicente, I. (2007). Pathways of river nutrients towards the euphotic zone in a deep-reservoir of small size: Uncertainty analysis. *Ecological Modelling*, 202(3–4), 345–361. https://doi.org/10.1016/j.ecolmodel.2006.11.006

Shi, H. (2025). Data set for “Buoyancy-tilted shear layers and coherent structures in hyperpycnal river plume plunging” [Dataset]. Zenodo. https://doi.org/10.5281/zenodo.15276522

Shi, H., Negretti, M. E., Chauchat, J., Blanckaert, K., Lemmin, U., & Barry, D. A. (2022). Unconfined plunging of a hyperpycnal river plume over a sloping bed and its lateral spreading: Laboratory experiments and numerical modeling. *Water Resources Research*, 58(8), e2022WR032633. https://doi.org/10.1029/2022WR032633

Shi, H., Negretti, M. E., Chauchat, J., Blanckaert, K., Lemmin, U., & Barry, D. A. (2024). Tracking the nearfield evolution of an initially shallow, neutrally buoyant plane jet over a sloping bottom boundary. *Water Resources Research*, 60(4), e2023WR034826. https://doi.org/10.1029/2023WR034826

Shi, H., Zhang, J., & Huai, W. (2023). Experimental study on velocity distributions, secondary currents, and coherent structures in open channel flow with submerged riparian vegetation. *Advances in Water Resources*, 173, 104406. https://doi.org/10.1016/j.advwatres.2023.104406

Soulignac, F., Lemmin, U., Ziabari, S. M. H., Wynn, H. K., Graf, B., & Barry, D. A. (2021). Rapid changes in river plume dynamics caused by advected wind-driven coastal upwelling as observed in Lake Geneva. *Limnology & Oceanography*, 66(8), 3116–3133. https://doi.org/10.1002/lno.11864

Spigel, R. H., Howard-Williams, C., Gibbs, M., Stephens, S., & Waugh, B. (2005). Field calibration of a formula for entrance mixing of river inflows to lakes: Lake Taupo, North Island, New Zealand. *New Zealand Journal of Marine & Freshwater Research*, 39(4), 785–802. https://doi.org/10.1080/00288330.2005.951735

Stocchino, A., & Brocchini, M. (2010). Horizontal mixing of quasi-uniform straight compound channel flows. *Journal of Fluid Mechanics*, 643, 425–435. https://doi.org/10.1017/S0022112009992680

Tennekes, H., & Lumley, J. L. (1972). *A first course in turbulence*. MIT press. Retrieved from https://mitpress.mit.edu/9780262536301/a-first-course-in-turbulence

Thorez, S., Blanckaert, K., Lemmin, U., & Barry, D. A. (2024). Quantifying mixing processes at a river-lake interface: The case of the plunging negatively buoyant inflow of the Rhône River into Lake Geneva. In *River Flow 2022* (pp. 892–899). CRC Press. https://doi.org/10.1201/9781003323037-120

Thorez, S., Lemmin, U., Barry, D. A., & Blanckaert, K. (2024). Hydro-sedimentary processes of a plunging hyperpycnal river plume revealed by synchronized remote imagery and gridded current measurements. *Water Resources Research*, 60(3), e2023WR035907. https://doi.org/10.1029/2023WR035907

Truong, S. H., & Uijttewaal, W. S. J. (2019). Transverse momentum exchange induced by large coherent structures in a vegetated compound channel. *Water Resources Research*, 55(1), 589–612. https://doi.org/10.1029/2018WR023273

Uijttewaal, W. S. J., & Booij, R. (2000). Effects of shallowness on the development of free-surface mixing layers. *Physics of Fluids*, 12(2), 392–402. https://doi.org/10.1063/1.870317

Vandekerkhove, E., Bertrand, S., Crescenzi Lanna, E., Reid, B., & Pantoja, S. (2020). Modern sedimentary processes at the heads of Martínez channel and Steffen Fjord, Chilean Patagonia. *Marine Geology*, 419, 106076. https://doi.org/10.1016/j.margeo.2019.106076

Variano, E. A., & Cowen, E. A. (2013). Turbulent transport of a high-Schmidt-number scalar near an air-water interface. *Journal of Fluid Mechanics*, 731, 259–287. https://doi.org/10.1017/jfm.2013.273

Wallace, J. M. (2016). Quadrant analysis in turbulence research: History and evolution. *Annual Review of Fluid Mechanics*, 48(1), 131–158. https://doi.org/10.1146/annurev-fluid-122414-034550

Weller, H. G., Tabor, G., Jasak, H., & Fureby, C. (1998). A tensorial approach to computational continuum mechanics using object-oriented techniques. *Computers in Physics*, 12(6), 620–631. https://doi.org/10.1063/1.168744

White, B. L., & Helfrich, K. R. (2013). Rapid gravitational adjustment of horizontal shear flows. *Journal of Fluid Mechanics*, 721, 86–117. https://doi.org/10.1017/jfm.2013.41

White, B. L., & Nepf, H. M. (2007). Shear instability and coherent structures in shallow flow adjacent to a porous layer. *Journal of Fluid Mechanics*, 593, 1–32. https://doi.org/10.1017/S0022112007008415

White, B. L., & Nepf, H. M. (2008). A vortex-based model of velocity and shear stress in a partially vegetated shallow channel. *Water Resources Research*, 44(1), W01412. https://doi.org/10.1029/2006WR005651

Yuan, Y., & Horner-Devine, A. R. (2017). Experimental investigation of large-scale vortices in a freely spreading gravity current. *Physics of Fluids*, 29(10), 106603. https://doi.org/10.1063/1.5006176