# Three Kinds of Butterfly Effects within Lorenz Models

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## Definition

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## 1. Introduction

## 2. Definitions of Butterfly Effects

#### 2.1. The First Kind of Butterfly Effect (BE1)

#### 2.2. The Second Kind of Butterfly Effect (BE2)

- Predictability; Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?
- In more technical language, is the behavior of the atmosphere unstable (“on all spatial scales”) with respect to perturbations of small amplitude?
- How can we determine whether the atmosphere is unstable?

- One hypothesis, unconfirmed, is that the influence of a butterfly’s wings will spread in turbulent air, but not in calm air;
- We must therefore leave our original question (i.e., the first question) unanswered for a few more years, even while affirming our faith in the instability of the atmosphere (i.e., the second and third questions).

#### 2.3. The Third Kind of Butterfly Effect (BE3)

It is proposed that certain formally deterministic fluid systems which possess many scales of motion are observationally indistinguishable from indeterministic systems; specifically that two states of the system differing initially by a small observational error will evolve into two states differing as greatly as randomly chosen states of the system within a finite time interval, which cannot be lengthened by reducing the amplitude of the initial error.

- 1.
- The turnover time (${\tau}_{k})$ is the time for a parcel with velocity ${v}_{k}$ to move a distance of $1/k$, with ${v}_{k}$ being the velocity associated with wavenumber $k$ (e.g., Vallis, 2006 [44]).
- 2.
- The saturation time (${t}_{k})$ is defined as the time for the perturbation at wavenumber $k$ to become saturated (i.e., reaching the value of background kinetic energy). In [3], the saturation time (${t}_{k}$) determines the predictability horizon at wavenumber $k.$

## 3. Discussion

#### 3.1. A Popular but Inaccurate Analogy for BE1 and Chaos

“For want of a nail, the shoe was lost.For want of a shoe, the horse was lost.For want of a horse, the rider was lost.For want of a rider, the battle was lost.For want of a battle, the kingdom was lost.And all for the want of a horseshoe nail”.

#### 3.2. A Positive Contribution by Small Scale Processes Within BE3

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The sensitive dependence on initial conditions (

**a**) and diverged trajectories (

**b**–

**d**) using the Lorenz 1963 model with $r=28$ and $\sigma =10$. Each panel displays solutions from the control (in blue) and parallel (in red) runs, respectively. The control run has an initial condition of (X, Y, Z) = (0, 1, 0), and the parallel run additionally includes a small perturbation (1 × 10

^{−10}) in the initial value of Y. Chaotic solutions in the X-Y-Z phase space within the Lorenz model can be found in Figure 2 of [16].

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**MDPI and ACS Style**

Shen, B.-W.; Pielke, R.A., Sr.; Zeng, X.; Cui, J.; Faghih-Naini, S.; Paxson, W.; Atlas, R.
Three Kinds of Butterfly Effects within Lorenz Models. *Encyclopedia* **2022**, *2*, 1250-1259.
https://doi.org/10.3390/encyclopedia2030084

**AMA Style**

Shen B-W, Pielke RA Sr., Zeng X, Cui J, Faghih-Naini S, Paxson W, Atlas R.
Three Kinds of Butterfly Effects within Lorenz Models. *Encyclopedia*. 2022; 2(3):1250-1259.
https://doi.org/10.3390/encyclopedia2030084

**Chicago/Turabian Style**

Shen, Bo-Wen, Roger A. Pielke, Sr., Xubin Zeng, Jialin Cui, Sara Faghih-Naini, Wei Paxson, and Robert Atlas.
2022. "Three Kinds of Butterfly Effects within Lorenz Models" *Encyclopedia* 2, no. 3: 1250-1259.
https://doi.org/10.3390/encyclopedia2030084