# Probability Distribution of Extreme Events in a Baroclinic Wave Laboratory Experiment

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## Abstract

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## 1. Introduction

## 2. Experimental Set-Up

## 3. Data

## 4. Results

#### 4.1. Extreme Value Distributions

`gevfit`with maximum likelihood estimation (for details, see, for example, [22,23]). We further show the confidence intervals of these parameters (${\xi}_{95\%}^{\pm}$, ${\sigma}_{95\%}^{\pm}$, ${\mu}_{95\%}^{\pm}$) at a 95% confidence level.

#### 4.2. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Domain of the working fluid chamber covered by the IR-camera (

**left**). The thick magenta line shows the azimuthal sections along which data were collected to obtain the Hovmoeller plot (

**right**). In this example, 1 h of data are shown. Waves propagate counterclockwise. Color code: dark blue = $22.5$ ${}^{\circ}$C; dark red = $24.4$ ${}^{\circ}$C.

**Figure 4.**The experimental block data of extremes of mean temperature ${E}_{ext}^{\prime}\left(j\right)$ (

**top**) in K and variance ${V}_{ext}^{\prime}\left(j\right)$ (

**bottom**) in K${}^{2}$.

**Figure 5.**The NCEP block data of extremes of mean temperature ${E}_{ext}^{\prime}\left(j\right)$ (

**top**) in K and variance ${V}_{ext}^{\prime}\left(j\right)$ (

**bottom**) in K${}^{2}$.

**Figure 6.**Empirical probability density distributions of ${E}_{ext}^{\prime}\left(j\right)$ and ${V}_{ext}^{\prime}\left(j\right)$ values. (

**a**) Experiment. (

**b**) NCEP data.

**Figure 7.**Distributions for parameters $\xi $, $\sigma $, and $\mu $ and all combinations of parameters ${\xi}_{95\%}^{\pm}$, ${\sigma}_{95\%}^{\pm}$, and ${\mu}_{95\%}^{\pm}$, as given in Table 3. Blue (red) curves show the experimental (NCEP) data.

**Table 1.**Parameters of the laboratory experiment. See also Figure 1 (left).

Geometry | |||
---|---|---|---|

inner radius | a (mm) | 350 | |

outer radius | b (mm) | 700 | |

gap width | b–a (mm) | 350 | |

fluid depth | d (mm) | 60 | |

Exp. Parameters | |||

temperature difference | $\Delta T$ (K) | 4.0 | |

revolution speed | $\mathsf{\Omega}$ (rpm) | 2.0 | |

Fluid Properties | |||

density | $\rho $ ($\mathrm{kg}\phantom{\rule{4.pt}{0ex}}{\mathrm{m}}^{-3}$) | 997 | |

kin. viscosity | $\nu $ (${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$) | 1.004 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | |

therm. conductivity | $\kappa $ (${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$) | 0.1434 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | |

exp. coefficient | $\alpha $ (1/K) | 0.207 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |

Similarity Parameters | |||

Prandtl number | Pr = $\nu /\kappa $ | 7.00 | |

Rossby number | Ro | 0.91 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |

Taylor number | Ta | 1.52 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{10}$ |

**Table 2.**Infrared camera’s technical specifications. The infrared camera used was from Jenoptik, module IR-TCM 640. The software was the IRBIS package developed by InfraTec GmbH, Dresden, Germany.

Image format | 1024 × 680 pixel |
---|---|

Spectral range | 7.5–14 $\mathsf{\mu}$m |

Range for measuring and visualization | $233.15$–$573.15$ K |

Thermal sensitivity | <80 mK |

Measurement accuracy | ±1.5 K |

Dynamic range | 16 bit |

Image rate | 60 Hz |

**Table 3.**Estimated parameters for the GEV distribution and the confidence intervals at a 95% confidence level.

$\mathit{\xi}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | ${\mathit{\xi}}_{95\%}^{\pm}$ | ${\mathit{\sigma}}_{95\%}^{\pm}$ | ${\mathit{\mu}}_{95\%}^{\pm}$ | |
---|---|---|---|---|---|---|

GEV${}_{exp}^{E}$ | 0.1230 | 0.1157 | 0.3621 | −0.0585 | 0.0967 | 0.3353 |

0.3044 | 0.1384 | 0.3889 | ||||

GEV${}_{exp}^{V}$ | 0.1136 | 0.1055 | 0.2903 | −0.0387 | 0.0890 | 0.2666 |

0.2660 | 0.1252 | 0.3140 | ||||

GEV${}_{ncep}^{E}$ | 0.1204 | 0.1161 | 0.2809 | −0.0800 | 0.0965 | 0.2536 |

0.3207 | 0.1396 | 0.3082 | ||||

GEV${}_{ncep}^{V}$ | 0.1996 | 0.1115 | 0.1870 | −0.0020 | 0.0922 | 0.1608 |

0.4012 | 0.1347 | 0.2131 |

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**MDPI and ACS Style**

Harlander, U.; Borcia, I.D.; Vincze, M.; Rodda, C.
Probability Distribution of Extreme Events in a Baroclinic Wave Laboratory Experiment. *Fluids* **2022**, *7*, 274.
https://doi.org/10.3390/fluids7080274

**AMA Style**

Harlander U, Borcia ID, Vincze M, Rodda C.
Probability Distribution of Extreme Events in a Baroclinic Wave Laboratory Experiment. *Fluids*. 2022; 7(8):274.
https://doi.org/10.3390/fluids7080274

**Chicago/Turabian Style**

Harlander, Uwe, Ion Dan Borcia, Miklos Vincze, and Costanza Rodda.
2022. "Probability Distribution of Extreme Events in a Baroclinic Wave Laboratory Experiment" *Fluids* 7, no. 8: 274.
https://doi.org/10.3390/fluids7080274