# Beyond Scale-by-Scale Equilibrium

## Abstract

**:**

## 1. Introduction

## 2. The Turbulence Problem

## 3. Turbulence Dissipation and Equilibrium Cascade

- (i)
- (ii)
- (iii)
- Predicting the power law decay of homogeneous isotropic turbulence (e.g., [39]);
- (iv)
- (v)
- (vi)

- (a)
- Local homogeneity and therefore zero average two-point turbulence production rate, zero average two-point interspace turbulence transport rate, and zero average two-point pressure gradient-velocity correlations at scale r;
- (b)
- Local stationarity and, therefore, zero average rate of change of the turbulent kinetic energy at scales smaller than r.

## 4. Non-Equilibrium Turbulence Dissipation Laws

## 5. Classification of Scale-by-Scale Non-Equilibria

#### 5.1. Category A: Statistically Stationary and Homogeneous Turbulence

#### 5.2. Category B: Non-Stationary Statistically Homogeneous Turbulence

#### 5.3. Category C: Non-Homogeneous Statistically Stationary Turbulence with Two-Point Turbulence Production and without Two-Point Interspace Transport

#### 5.4. Category D: Non-Homogeneous Statistically Stationary Turbulence with Negligible Two-Point Turbulence Production

#### 5.5. Category E: Streamwise-Decaying Non-Homogeneous Turbulence with Negligible Two-Point Turbulence Production

#### 5.6. Category F: External Intermittency: A Case of Extreme Fluctuating Non-Homogeneity and Non-Stationarity

## 6. Conclusions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Vassilicos, J.C.
Beyond Scale-by-Scale Equilibrium. *Atmosphere* **2023**, *14*, 736.
https://doi.org/10.3390/atmos14040736

**AMA Style**

Vassilicos JC.
Beyond Scale-by-Scale Equilibrium. *Atmosphere*. 2023; 14(4):736.
https://doi.org/10.3390/atmos14040736

**Chicago/Turabian Style**

Vassilicos, John C.
2023. "Beyond Scale-by-Scale Equilibrium" *Atmosphere* 14, no. 4: 736.
https://doi.org/10.3390/atmos14040736