# Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Two-Dimensional Turbulence

#### 2.2. Lévy Bases

- (a)
- $L\left(A\right)$ is infinitely divisible for all $A\in {B}_{b}\left(S\right)$;
- (b)
- $L\left({A}_{1}\right),\dots ,L\left({A}_{n}\right)$ are independent for disjoint subsets ${A}_{1},\dots ,{A}_{n}\in {B}_{b}\left(S\right)$;
- (c)
- For disjoint subsets ${A}_{1},{A}_{2},\dots \in {B}_{b}\left(S\right)$ with ${\cup}_{i=1}^{\infty}{A}_{i}\in {B}_{b}\left(S\right)$ we have$$L\left({\cup}_{i=1}^{\infty}{A}_{i}\right)=\sum _{i=1}^{\infty}L\left({A}_{i}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a.s.$$

#### 2.3. Ambit Fields

## 3. Modeling Framework

#### 3.1. Stream Function

#### 3.2. Vector Field

## 4. Skewness

#### 4.1. Decomposition

**Theorem**

**1.**

#### 4.2. Triads

**Definition**

**1.**

**Theorem**

**2.**

## 5. Key Example

## 6. Discussion

## Funding

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Theorem 1.**

## References

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**Figure 2.**Third-order structure function ${S}_{3}\left(x\right)$ as a function of x in double-logarithmic representation for the kernel (17). The straight line for small x indicates a scaling behavior $\propto {x}^{3}$ and the straight line for large x indicates a linear behavior $\propto x$.

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**MDPI and ACS Style**

Schmiegel, J.
Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions. *Symmetry* **2020**, *12*, 1265.
https://doi.org/10.3390/sym12081265

**AMA Style**

Schmiegel J.
Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions. *Symmetry*. 2020; 12(8):1265.
https://doi.org/10.3390/sym12081265

**Chicago/Turabian Style**

Schmiegel, Jürgen.
2020. "Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions" *Symmetry* 12, no. 8: 1265.
https://doi.org/10.3390/sym12081265