# Periodic Flows in a Viscous Stratified Fluid in a Homogeneous Gravitational Field

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## Abstract

**:**

## 1. Introduction

## 2. System of Fundamental Equations of Periodic Flows in the Atmosphere and Ocean

#### 2.1. The Complete System of Equations Determining the Flow of the Liquid

#### 2.2. The Reduced System of Equations

## 3. Periodic Flows in the Thickness of a Uniformly Stratified Liquid

#### 3.1. Linearization of the Equation System

#### 3.2. Dispersion Relation: Classification of Flow Components

## 4. High-Frequency Acoustic Waves

## 5. Low-Frequency Gravity Waves

## 6. Periodic Flows in a Two-Layer System of Stratified Liquids

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Parameter | Fluid | |||
---|---|---|---|---|

Stratified | Homogeneous | |||

Strongly | Weakly | Potentially | Actually | |

Buoyancy frequency $N,{s}^{-1}$ | 1 | 0.01 | 0.00001 | 0.0 |

Buoyancy period ${T}_{b}$ | 10 s | 10 min | 10 days | $\infty $ |

Scale of stratification $\Lambda $ | 10 m | 100 km | ${10}^{8}$ km | $\infty $ |

Viscous wave scale ${\delta}_{N}^{g\nu}={\left(g\nu \right)}^{1/3}{N}^{-1}$, cm | 2 | 200 | $2\cdot {10}^{5}$ | $\infty $ |

Stokes microscale ${\delta}_{N}^{\nu}=\sqrt{\nu /N}$, cm | 0.1 | 1 | 30 | $\infty $ |

Parameter | Fluid | |||
---|---|---|---|---|

Stratified | Homogeneous | |||

Strongly | Weakly | Potentially | Actually | |

Buoyancy frequency $N,{s}^{-1}$ | 1 | 0.01 | 0.00001 | 0.0 |

Buoyancy period ${T}_{b}$ | 10 s | 10 min | 10 days | $\infty $ |

Scale of stratification $\Lambda $ | 10 m | 100 km | ${10}^{8}$ km | $\infty $ |

Viscous wave scale ${\delta}_{N}^{g\nu}={\left(g\nu \right)}^{1/3}{N}^{-1}$, cm | 5 | 500 | $5\cdot {10}^{5}$ | $\infty $ |

Stokes microscale ${\delta}_{N}^{\nu}=\sqrt{\nu /N}$, cm | 0.4 | 4 | 120 | $\infty $ |

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**MDPI and ACS Style**

Chashechkin, Y.D.; Ochirov, A.A.
Periodic Flows in a Viscous Stratified Fluid in a Homogeneous Gravitational Field. *Mathematics* **2023**, *11*, 4443.
https://doi.org/10.3390/math11214443

**AMA Style**

Chashechkin YD, Ochirov AA.
Periodic Flows in a Viscous Stratified Fluid in a Homogeneous Gravitational Field. *Mathematics*. 2023; 11(21):4443.
https://doi.org/10.3390/math11214443

**Chicago/Turabian Style**

Chashechkin, Yuli D., and Artem A. Ochirov.
2023. "Periodic Flows in a Viscous Stratified Fluid in a Homogeneous Gravitational Field" *Mathematics* 11, no. 21: 4443.
https://doi.org/10.3390/math11214443