# Turbulence as a Network of Fourier Modes

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## Abstract

**:**

## 1. Introduction

## 2. Fourier Space Formulation

- a finite number of nodes in $\mathbf{k}$ space (the grid in the above example);
- a list of node to pair couplings ${\mathbf{i}}_{\ell}$ to be determined by the triadic interaction condition;
- the coefficient of interaction ${M}_{\ell ,{\ell}^{\prime},{\ell}^{\u2033}}^{ij\kappa}$ for each of these couplings; and
- a set of field variables (e.g., ${u}_{\ell}^{x},{u}_{\ell}^{y},{u}_{\ell}^{z}$) to evolve on each node of the network using Equation (5).

#### 2.1. Energy Transfer

#### 2.2. Network Reduction

#### 2.3. Transfer Rates

## 3. Spectral Reduction

#### 3.1. Phase Dynamics: Synchronization vs. Random Phase

#### 3.2. Beyond Spectral Reduction

## 4. Examples of Network Models

#### 4.1. Nested Polyhedra Models

#### 4.2. Predator–Prey Models

#### 4.3. Food Webs

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The standard GOY model on the left (10 nodes, shown as an unclosed ring), and the small world version on the right, with some randomly added triads. Note that each triad enters an additional link to the list of connections for each of its three nodes. Note that the triangles for these disparate scale interactions are very elongated.

**Table 1.**$\ell =8m+{n}_{m}$, where ${n}_{m}$, which denotes the nth vertex of the mth polyhedron, is shown on the leftmost column, interacts with ${\mathbf{i}}_{\ell}=\left\{{\ell}^{\prime},{\ell}^{\u2033}\right\}=\left\{8m-16+{n}_{m-2},8m-10+{n}_{m-1}\right\}$, $\left\{8m-10+{n}_{m-1},8m+6+{n}_{m+1}\right\}$ and $\left\{8m+6+{n}_{m+1},8m+16+{n}_{m+2}\right\}$ for an even m (i.e., an icosahedron node) where ${n}_{m}$, ${n}_{m\pm 1}$ and ${n}_{m\pm 2}$ are to be taken from their corresponding columns. Note that a bar over the integer value ${\ell}^{\prime}$ indicates ${c}_{{\ell}^{\prime}}=0$ (i.e., not conjugated), whereas no bar means ${c}_{{\ell}^{\prime}}=1$ (i.e., conjugated).

${\mathit{n}}_{\mathit{m}}$ | ${\mathit{i}}_{\mathit{n},\mathit{m}}=\left\{{\mathit{n}}_{\mathit{m}-2},{\mathit{n}}_{\mathit{m}-1}\right\}$ | ${\mathit{i}}_{\mathit{n},\mathit{m}}=\left\{{\mathit{n}}_{\mathit{m}-1},{\mathit{n}}_{\mathit{m}+1}\right\}$ | ${\mathit{i}}_{\mathit{n},\mathit{m}}=\left\{{\mathit{n}}_{\mathit{m}+1},{\mathit{n}}_{\mathit{m}+2}\right\}$ |
---|---|---|---|

0 | $\left\{\left(\overline{4},\overline{0}\right),\left(\overline{5},\overline{1}\right),\left(\overline{1},\overline{2}\right),\left(\overline{2},\overline{3}\right),\left(\overline{3},\overline{4}\right)\right\}$ | $\left\{\left(5,\overline{0}\right),\left(6,\overline{1}\right),\left(7,\overline{2}\right),\left(8,\overline{3}\right),\left(9,\overline{4}\right)\right\}$ | $\left\{\left(\overline{5},\overline{4}\right),\left(\overline{6},\overline{5}\right),\left(\overline{7},\overline{1}\right),\left(\overline{8},\overline{2}\right),\left(\overline{9},\overline{3}\right)\right\}$ |

1 | $\left\{\left(3,\overline{0}\right),\left(4,\overline{4}\right),\left(\overline{5},\overline{5}\right),\left(\overline{0},7\right),\left(\overline{2},\overline{9}\right)\right\}$ | $\left\{\left(1,\overline{0}\right),\left(3,\overline{4}\right),\left(\overline{8},\overline{5}\right),\left(\overline{2},7\right),\left(\overline{6},\overline{9}\right)\right\}$ | $\left\{\left(\overline{1},3\right),\left(\overline{3},4\right),\left(8,\overline{5}\right),\left(2,\overline{0}\right),\left(6,\overline{2}\right)\right\}$ |

2 | $\left\{\left(5,\overline{0}\right),\left(4,\overline{1}\right),\left(\overline{3},\overline{5}\right),\left(\overline{1},\overline{6}\right),\left(\overline{0},8\right)\right\}$ | $\left\{\left(4,\overline{0}\right),\left(2,\overline{1}\right),\left(\overline{7},\overline{5}\right),\left(\overline{9},\overline{6}\right),\left(\overline{3},8\right)\right\}$ | $\left\{\left(\overline{4},5\right),\left(\overline{2},4\right),\left(7,\overline{3}\right),\left(9,\overline{1}\right),\left(3,\overline{0}\right)\right\}$ |

3 | $\left\{\left(1,\overline{1}\right),\left(5,\overline{2}\right),\left(\overline{4},\overline{6}\right),\left(\overline{2},\overline{7}\right),\left(\overline{0},9\right)\right\}$ | $\left\{\left(0,\overline{1}\right),\left(3,\overline{2}\right),\left(\overline{8},\overline{6}\right),\left(\overline{5},\overline{7}\right),\left(\overline{4},9\right)\right\}$ | $\left\{\left(\overline{0},1\right),\left(\overline{3},5\right),\left(8,\overline{4}\right),\left(5,\overline{2}\right),\left(4,\overline{0}\right)\right\}$ |

4 | $\left\{\left(\overline{0},5\right),\left(1,\overline{3}\right),\left(2,\overline{2}\right),\left(\overline{3},\overline{8}\right),\left(\overline{5},\overline{7}\right)\right\}$ | $\left\{\left(\overline{0},5\right),\left(4,\overline{3}\right),\left(1,\overline{2}\right),\left(\overline{6},\overline{8}\right),\left(\overline{9},\overline{7}\right)\right\}$ | $\left\{\left(0,\overline{0}\right),\left(\overline{4},1\right),\left(\overline{1},2\right),\left(6,\overline{3}\right),\left(9,\overline{5}\right)\right\}$ |

5 | $\left\{\left(\overline{0},6\right),\left(\overline{1},\overline{8}\right),\left(2,\overline{4}\right),\left(3,\overline{3}\right),\left(\overline{4},\overline{9}\right)\right\}$ | $\left\{\left(\overline{1},6\right),\left(\overline{5},\overline{8}\right),\left(0,\overline{4}\right),\left(2,\overline{3}\right),\left(\overline{7},\overline{9}\right)\right\}$ | $\left\{\left(1,\overline{0}\right),\left(5,\overline{1}\right),\left(\overline{0},2\right),\left(\overline{2},3\right),\left(7,\overline{4}\right)\right\}$ |

**Table 2.**$\ell =8m+{n}_{m}+2$ where ${n}_{m}$ is shown on the leftmost column, interacts with ${\mathbf{i}}_{\ell}=\left\{{\ell}^{\prime},{\ell}^{\u2033}\right\}=\left\{8m-14+{n}_{m-2},8m-4+{n}_{m-1}\right\}$, $\left\{8m-4+{n}_{m-1},8m+12+{n}_{m+1}\right\}$ and $\left\{8m+12+{n}_{m+1},8m+18+{n}_{m+1}\right\}$ for an odd m (i.e., a dodecahedron node) where ${n}_{m}$, ${n}_{m\pm 1}$ and ${n}_{m\pm 2}$ are to be taken from their corresponding columns. As in Table 1, if the integer value ${\ell}^{\prime}$ has a bar over it ${c}_{{\ell}^{\prime}}=0$ (i.e., not conjugated), whereas no bar means ${c}_{{\ell}^{\prime}}=1$ (i.e., conjugated).

${\mathit{n}}_{\mathit{m}}$ | ${\mathit{i}}_{\mathit{n},\mathit{m}}=\left\{{\mathit{n}}_{\mathit{m}-2},{\mathit{n}}_{\mathit{m}-1}\right\}$ | ${\mathit{i}}_{\mathit{n},\mathit{m}}=\left\{{\mathit{n}}_{\mathit{m}-1},{\mathit{n}}_{\mathit{m}+1}\right\}$ | ${\mathit{i}}_{\mathit{n},\mathit{m}}=\left\{{\mathit{n}}_{\mathit{m}+1},{\mathit{n}}_{\mathit{m}+2}\right\}$ |
---|---|---|---|

0 | $\left\{\left(\overline{5},\overline{0}\right),\left(\overline{1},\overline{1}\right),\left(\overline{4},\overline{2}\right)\right\}$ | $\left\{\left(4,\overline{0}\right),\left(\overline{3},\overline{1}\right),\left(\overline{5},\overline{2}\right)\right\}$ | $\left\{\left(\overline{4},\overline{5}\right),\left(3,\overline{1}\right),\left(5,\overline{4}\right)\right\}$ |

1 | $\left\{\left(\overline{6},\overline{0}\right),\left(\overline{2},\overline{2}\right),\left(\overline{0},\overline{3}\right)\right\}$ | $\left\{\left(5,\overline{0}\right),\left(\overline{4},\overline{2}\right),\left(\overline{1},\overline{3}\right)\right\}$ | $\left\{\left(\overline{5},\overline{6}\right),\left(4,\overline{2}\right),\left(1,\overline{0}\right)\right\}$ |

2 | $\left\{\left(\overline{7},\overline{0}\right),\left(\overline{3},\overline{3}\right),\left(\overline{1},\overline{4}\right)\right\}$ | $\left\{\left(1,\overline{0}\right),\left(\overline{5},\overline{3}\right),\left(\overline{2},\overline{4}\right)\right\}$ | $\left\{\left(\overline{1},\overline{7}\right),\left(5,\overline{3}\right),\left(2,\overline{1}\right)\right\}$ |

3 | $\left\{\left(\overline{8},\overline{0}\right),\left(\overline{4},\overline{4}\right),\left(\overline{2},\overline{5}\right)\right\}$ | $\left\{\left(2,\overline{0}\right),\left(\overline{1},\overline{4}\right),\left(\overline{3},\overline{5}\right)\right\}$ | $\left\{\left(\overline{2},\overline{8}\right),\left(1,\overline{4}\right),\left(3,\overline{2}\right)\right\}$ |

4 | $\left\{\left(\overline{9},\overline{0}\right),\left(\overline{3},\overline{1}\right),\left(\overline{0},\overline{5}\right)\right\}$ | $\left\{\left(3,\overline{0}\right),\left(\overline{4},\overline{1}\right),\left(\overline{2},\overline{5}\right)\right\}$ | $\left\{\left(\overline{3},\overline{9}\right),\left(4,\overline{3}\right),\left(2,\overline{0}\right)\right\}$ |

5 | $\left\{\left(8,\overline{1}\right),\left(7,\overline{2}\right),\left(\overline{0},4\right)\right\}$ | $\left\{\left(5,\overline{1}\right),\left(3,\overline{2}\right),\left(\overline{0},4\right)\right\}$ | $\left\{\left(\overline{5},8\right),\left(\overline{3},7\right),\left(0,\overline{0}\right)\right\}$ |

6 | $\left\{\left(9,\overline{2}\right),\left(8,\overline{3}\right),\left(1,5\right)\right\}$ | $\left\{\left(1,\overline{2}\right),\left(4,\overline{3}\right),\left(\overline{0},5\right)\right\}$ | $\left\{\left(\overline{1},9\right),\left(\overline{4},8\right),\left(0,\overline{1}\right)\right\}$ |

7 | $\left\{\left(\overline{2},1\right),\left(5,\overline{3}\right),\left(9,\overline{4}\right)\right\}$ | $\left\{\left(\overline{0},1\right),\left(2,\overline{3}\right),\left(5,\overline{4}\right)\right\}$ | $\left\{\left(0,\overline{2}\right),\left(\overline{2},5\right),\left(\overline{5},9\right)\right\}$ |

8 | $\left\{\left(\overline{3},2\right),\left(6,\overline{4}\right),\left(5,\overline{5}\right)\right\}$ | $\left\{\left(\overline{0},2\right),\left(3,\overline{4}\right),\left(1,\overline{5}\right)\right\}$ | $\left\{\left(0,\overline{3}\right),\left(\overline{3},6\right),\left(\overline{1},5\right)\right\}$ |

9 | $\left\{\left(6,\overline{1}\right),\left(\overline{4},3\right),\left(7,\overline{5}\right)\right\}$ | $\left\{\left(2,\overline{1}\right),\left(\overline{0},3\right),\left(4,\overline{5}\right)\right\}$ | $\left\{\left(\overline{2},6\right),\left(0,\overline{4}\right),\left(\overline{4},7\right)\right\}$ |

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Gürcan, Ö.D.; Li, Y.; Morel, P.
Turbulence as a Network of Fourier Modes. *Mathematics* **2020**, *8*, 530.
https://doi.org/10.3390/math8040530

**AMA Style**

Gürcan ÖD, Li Y, Morel P.
Turbulence as a Network of Fourier Modes. *Mathematics*. 2020; 8(4):530.
https://doi.org/10.3390/math8040530

**Chicago/Turabian Style**

Gürcan, Özgür. D., Yang Li, and Pierre Morel.
2020. "Turbulence as a Network of Fourier Modes" *Mathematics* 8, no. 4: 530.
https://doi.org/10.3390/math8040530