# Potential Vorticity Generation in Breaking Gravity Waves

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Equations

#### 2.2. Scale Analysis

#### 2.3. Numerical Approach

## 3. Results

#### 3.1. Main Simulation

#### 3.2. Sensitivity to Reynolds and Froude Numbers

#### 3.3. Sensitivity to Numerical Resolution

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Time series from the Main simulation of (

**a**) kinetic and potential energy, (

**b**) kinetic and potential energy dissipation, (

**c**) turbulent Froude number, and (

**d**) potential enstrophy. The inset in (

**d**) shows potential enstrophy on a log scale to illustrate the decay of the initial potential enstrophy from random noise.

**Figure 2.**Vertical $(x,z)$ slices of (

**a**–

**e**) ${b}_{\mathrm{tot}}$, (

**f**–

**j**) ${\omega}_{y}$, and (

**k**–

**o**) q at (from top to bottom) $t=16$, 20, 24, 28, and 36 from Main simulation. All slices are shown at $y=0$.

**Figure 3.**Time series of the terms in the potential enstrophy Equation (8) from the Main simulation.

**Figure 4.**(

**a**) Vertical $(x,z)$ slice of q, and horizontal $(x,y)$ slices of b through (

**b**) $z={z}_{1}$ and (

**c**) $z={z}_{2}$, for the Main simulation at $t=24$. Note that ${z}_{1}$ and ${z}_{2}$ are shown in (

**a**). The vertical line segments in (

**a**) mark the locations of significant PV along $z={z}_{1}$ and $z={z}_{2}$; these locations are marked with ticks in (

**b**) for ${z}_{1}$ and (

**c**) for ${z}_{2}$, respectively.

**Figure 5.**Energy and potential enstrophy spectra from the Main simulation at $t=28$. The Ozmidov and Kolmogorov wavenumbers ${k}_{O}=1/{L}_{O}$ and ${k}_{d}=1/\eta $ are shown.

**Figure 6.**Time series of potential enstrophy from simulations with (

**a**) different $Re$ and (

**b**) different $Fr$. A log scale is used in (

**a**) with a linear scale in the inset.

**Figure 7.**Potential enstrophy spectra from simulations with (

**a**) different $Re$ and (

**b**) different $Fr$, at the time of maximum potential enstrophy. The vertical line segments mark the Kolmogorov wavenumbers in (

**a**) and the Ozmidov wavenumbers in (

**b**).

**Figure 8.**Maximum potential enstrophy V plotted against ${N}^{2}{\u03f5}^{2}/{\nu}^{2}$, where the maximum value of $\u03f5$ is used, for the Main simulation (black), simulations with different stratifications (High Strat and Higher Strat, in blue), and simulations with different viscosities (Low Visc and High Visc, in red). The reference lines are $V={N}^{2}{\u03f5}^{2}/{\nu}^{2}$ (solid) and $V=0.01{N}^{2}{\u03f5}^{2}/{\nu}^{2}$ (dashed).

Run | $\mathit{Fr}$ | $\sqrt{2}\mathit{a}/\mathit{N}$ | $\mathit{Re}$ | n | Max $\mathit{\u03f5}$ | $\mathit{\eta}{\mathit{k}}_{\mathit{T}}$ | Max ${\mathit{Re}}_{\mathit{b}}$ |
---|---|---|---|---|---|---|---|

Main | 1 | $\sqrt{2}$ | 3333 | 1024 | 0.0037 | 3.2 | 12.2 |

High Visc | 1 | $\sqrt{2}$ | 1667 | 512 | 0.0047 | 2.5 | 7.9 |

Low Visc | 1 | $\sqrt{2}$ | 4500 | 1728 | 0.0060 | 3.8 | 26.9 |

High Strat | $1/\sqrt{2}$ | 1 | 3333 | 1024 | 0.0052 | 2.9 | 8.6 |

Higher Strat | 1/2 | $1/\sqrt{2}$ | 3333 | 1034 | 0.0030 | 3.3 | 2.5 |

Low Res | 1 | $\sqrt{2}$ | 3333 | 512 | 0.0041 | 1.5 | 13.8 |

High Res | 1 | $\sqrt{2}$ | 3333 | 1536 | 0.0040 | 4.6 | 13.2 |

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**MDPI and ACS Style**

Waite, M.L.; Richardson, N.
Potential Vorticity Generation in Breaking Gravity Waves. *Atmosphere* **2023**, *14*, 881.
https://doi.org/10.3390/atmos14050881

**AMA Style**

Waite ML, Richardson N.
Potential Vorticity Generation in Breaking Gravity Waves. *Atmosphere*. 2023; 14(5):881.
https://doi.org/10.3390/atmos14050881

**Chicago/Turabian Style**

Waite, Michael L., and Nicholas Richardson.
2023. "Potential Vorticity Generation in Breaking Gravity Waves" *Atmosphere* 14, no. 5: 881.
https://doi.org/10.3390/atmos14050881