# Density Operator Approach to Turbulent Flows in Plasma and Atmospheric Fluids

## Abstract

**:**

## 1. Introduction

## 2. Hasegawa–Mima Model

#### 2.1. Hasegawa–Mima Equation for Plasma

#### 2.2. Fluctuations

#### 2.3. Observables

## 3. Wave-Mechanical Analogy in Zonal Flows

## 4. Density Operator Formalism

#### 4.1. Master Equation: Non-Sustainable Evolution

#### 4.2. Master Equation: Sustainable Evolution

#### 4.3. Time Evolution of Averages

## 5. Phase-Space Formulation: Non-Sustainable Evolution

#### 5.1. Wigner–Weyl Transform

#### 5.2. Eikonal Approximation

- Overall conservation occurs when$${\mu}_{\mathrm{dw}}^{\left(0\right)}={\mu}_{\mathrm{zf}}^{\left(0\right)}=0.$$In this case, both energy and enstrophy can flow from the drift-wave component to the zonal-flow one, and back, but do not leave the system:$$\begin{array}{ccc}\hfill {\mathcal{Z}}_{\mathrm{tot}}\left(t\right)& =& {\mathcal{Z}}_{\mathrm{tot}}\left(0\right)=\mathrm{const},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mathcal{E}}_{\mathrm{tot}}\left(t\right)& =& {\mathcal{E}}_{\mathrm{tot}}\left(0\right)=\mathrm{const}.\hfill \end{array}$$
- Overall exponential gain or loss occurs when$${\mu}_{\mathrm{dw}}^{\left(0\right)}={\mu}_{\mathrm{zf}}^{\left(0\right)}=\lambda /2,$$$$\begin{array}{ccc}\hfill {\mathcal{Z}}_{\mathrm{tot}}\left(t\right)& =& {\mathcal{Z}}_{\mathrm{tot}}\left(0\right)exp(-\lambda t),\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mathcal{E}}_{\mathrm{tot}}\left(t\right)& =& {\mathcal{E}}_{\mathrm{tot}}\left(0\right)exp(-\lambda t),\hfill \end{array}$$

## 6. Phase-Space Formulation: Sustainable Evolution

#### 6.1. Wigner–Weyl Transform

#### 6.2. Eikonal Approximation

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DW | Drift wave, drifton |

HME | Hasegawa–Mima equation |

NH | Non-Hermitian Hamiltonian |

RW | Rossby wave |

WKB | Wentzel–Kramers–Brillouin |

WKE | Wave kinetic equation |

ZF | Zonal flow |

## Appendix A. Derivation of Vorticity

## Appendix B. Wigner–Weyl Formalism

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Zloshchastiev, K.G.
Density Operator Approach to Turbulent Flows in Plasma and Atmospheric Fluids. *Universe* **2020**, *6*, 216.
https://doi.org/10.3390/universe6110216

**AMA Style**

Zloshchastiev KG.
Density Operator Approach to Turbulent Flows in Plasma and Atmospheric Fluids. *Universe*. 2020; 6(11):216.
https://doi.org/10.3390/universe6110216

**Chicago/Turabian Style**

Zloshchastiev, Konstantin G.
2020. "Density Operator Approach to Turbulent Flows in Plasma and Atmospheric Fluids" *Universe* 6, no. 11: 216.
https://doi.org/10.3390/universe6110216