# Vertical Shear Processes in River Plumes: Instabilities and Turbulent Mixing

^{*}

*Symmetry*)

## Abstract

**:**

## 1. Introduction

- What is the evolution of a sheared river plume base under different flow parameters?
- Which type of vertical shear instabilities can affect the plume base and under which flow conditions?
- What are the effects of the vertical shear instabilities on vertical turbulent mixing?

## 2. Materials and Methods

#### 2.1. Model Configuration

#### 2.2. Model Experiments

#### 2.3. Diagnostics for Stratified Sheared Flows Structure and Dynamics

#### 2.4. Diagnostics for Instabilities

#### 2.5. Diagnostics for Turbulent Mixing

## 3. Results

#### 3.1. Structure and Dynamics of Stratified Sheared Flows

#### 3.1.1. Reference Configuration

#### 3.1.2. Sensitivity to the Vertical Shear (Exp2)

#### 3.1.3. Sensitivity to the Thickness of the Interface (Exp3)

#### 3.1.4. Sensitivity to Topographic Ridges (Exp4)

#### 3.2. Modal Analysis and Instability Growth

#### 3.2.1. Reference Configuration

#### 3.2.2. Sensitivity to the Vertical Shear (Exp2)

#### 3.2.3. Sensitivity to the Thickness of the Interface (Exp3)

#### 3.2.4. Sensitivity to Topographic Ridges (Exp4)

#### 3.3. Turbulent Mixing

#### 3.3.1. Reference Configuration

#### 3.3.2. Sensitivity to the Vertical Shear (Exp2)

#### 3.3.3. Sensitivity to the Thickness of the Interface (Exp3)

#### 3.3.4. Sensitivity to Topographic Ridges (Exp4)

## 4. Discussion

#### 4.1. The Structure and Dynamics of Stratified Sheared Flows

#### 4.2. Development and Role of KH and Holmboe Instabilities

#### 4.3. Turbulent Mixing: Intensity and Efficiency

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Linear Stability Analysis

## Appendix B. Identification of the Holmboe Mode: The Bifurcation Theory

**Figure A1.**The bifurcation theory for the reference simulation (

**a**), experiment 2 (

**b**), experiment 3 (

**c**), and experiment 4 (

**d**). The blue solid vertical line indicates the identified Holmboe mode.

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**Figure 1.**The initial conditions for all the experiments: (

**top row**)–streamwise velocity; (

**bottom row**)—buoyancy.

**Figure 2.**Snapshots from the reference experiment of: buoyancy field (

**top two rows**), vorticity field (

**bottom two rows**). The black dots in the vorticity field indicate the vorticity centroid of each billow.

**Figure 3.**(

**a**) Snapshots of the Okubo–Weiss parameter in the reference experiment. (

**b**) Time evolution (reference experiment) of the aspect ratio of the two billows (

**top row**), the intensity of the strain (

**second row**), and its direction (

**bottom row**).

**Figure 4.**As Figure 2 for experiment 2.

**Figure 5.**As Figure 3 for experiment 2.

**Figure 6.**As Figure 2 for experiment 3.

**Figure 7.**As Figure 3 for experiment 3.

**Figure 8.**As Figure 2 for experiment 4.

**Figure 9.**As Figure 3 for experiment 4.

**Figure 10.**(

**a**) The growth rate, (

**b**) the vertical magnitude of the streamwise velocity for the Kelvin–Helmholtz mode, and (

**c**) the vertical magnitude of the streamwise velocity for the Holmboe mode for the reference and sensitivty experiments.

**Figure 11.**The total and modal decompositions of the EKE for the reference configuration (

**a**), experiment 2 (

**b**), experiment 3 (

**c**), and experiment 4 (

**d**).

**Figure 12.**Vertical eddy viscosity (black solid line), vertical diffusivity (black dashed line), and vertical dissipation of EKE (black solid line) and buoyancy (black dashed line) for the reference configuration (

**a**), experiment 2 (

**b**), experiment 3 (

**c**), and experiment 4 (

**d**).

**Figure 13.**Statistics and error bars (vertical segments at the end of the colored lines) for all experiments: (

**a**) vertical eddy viscosity, (

**b**) vertical diffusivity, (

**c**) vertical EKE dissipation, (

**d**) vertical buoyancy dissipation, (

**e**) turbulent Prandtl number, and (

**f**) irreversible mixing efficiency for KH (red) and Holmboe (green) instabilities.

Experiments | Initial Shear (s${}^{-2}$) | Initial Stratification (s${}^{-2}$) | Minimum Richardson Number | Interface | Bottom | Boundaries |
---|---|---|---|---|---|---|

Reference | 0.25 | 0.01 | 0.04 | plane (${h}_{b}={h}_{s}$) | flat | Periodic in x + Rigid in z |

Exp 2 | 0.5 | 0.01 | 0.02 | plane (${h}_{b}={h}_{s}$) | flat | Periodic in x + Rigid in z |

Exp 3 | 0.25 | 0.01 | 0.04 | plane (${h}_{b}=\frac{{h}_{s}}{1000}$) | flat | Periodic in x + Rigid in z |

Exp 4 | 0.25 | 0.01 | 0.04 | plane (${h}_{b}={h}_{s}$) | sloping | Periodic in x + Rigid in z |

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**MDPI and ACS Style**

Ayouche, A.; Carton, X.; Charria, G.
Vertical Shear Processes in River Plumes: Instabilities and Turbulent Mixing. *Symmetry* **2022**, *14*, 217.
https://doi.org/10.3390/sym14020217

**AMA Style**

Ayouche A, Carton X, Charria G.
Vertical Shear Processes in River Plumes: Instabilities and Turbulent Mixing. *Symmetry*. 2022; 14(2):217.
https://doi.org/10.3390/sym14020217

**Chicago/Turabian Style**

Ayouche, Adam, Xavier Carton, and Guillaume Charria.
2022. "Vertical Shear Processes in River Plumes: Instabilities and Turbulent Mixing" *Symmetry* 14, no. 2: 217.
https://doi.org/10.3390/sym14020217