# A Statistical Investigation of Mesoscale Precursors of Significant Tornadoes: The Italian Case Study

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Records of Tornadoes

#### 2.2. Reanalysis Datasets

#### 2.3. Meteorological Parameters

**Wind shear (LLS, DLS).**WS is computed as the magnitude of the vector representing the difference between the wind near the surface and at a fixed level in the air column. Two parameters are considered, the low-level shear (LLS) and the deep-level shear (DLS), representing the differences between the wind speed near the surface (1000 hPa) and at about 1 km and 6 km (900 hPa and 500 hPa, respectively). Although LLS and DLS are not calculated within the generally used 0–1km and 0–6 km layers, our different choice does not affect the results, also considering that all the analyzed tornadoes occur in flat regions (only one case is recorded above 300 m). It is important to highilight that previous studies [48,49] reveal that atmospheric reanalysis tends to underestimate the low-level shear compared to sounding observations.**Updraft Maximum Vertical Velocity (WM).**CAPE [50,51,52] represents the available potential energy. It is defined as $\mathrm{CAPE}=g{\int}_{LFC}^{EL}B\phantom{\rule{0.166667em}{0ex}}dz$, where z is the height, $B=g\frac{{T}_{Vp}-{T}_{Ve}}{{T}_{Ve}}$ is the buoyancy force, $LFC$ is the Level of Free Convection, $EL$ is the Equilibrium Level, and ${T}_{Vp}$ and ${T}_{Ve}$ are the virtual temperature of the air parcel and of the surrounding environment, respectively [23,53].The buoyancy produces the vertical acceleration of the air parcels: $B=dw/dt$, where w is the vertical velocity component and t is time. In the reanalyses used in this work, CAPE is extracted from ECMWF, calculated as the maximum value among the air parcels lifted from different model levels below 350 hPa (MUCAPE). In this study, CAPE is expressed via the Updraft Maximum Vertical Velocity (WM), supposing that the buoyancy is the only active force according to the Parcel Theory: i.e., $WM=\sqrt{2\xb7\mathrm{CAPE}}$ [23,53]. WM is computed at the same grid points and times as WS01 and WS06.**Standardized values.**To obtain comparable dimensionless indices, a standardization is applied. The generic index I is defined as $I=\frac{X-\overline{x}}{\sigma}$, where X is the value of the parameter of interest (WM, LLS, or DLS). The mean value $\overline{x}$ (climatological mean), as well as the standard deviation $\sigma $, are computed separately for each tornado, considering the 19 values of the parameter I at the same hour and calendar day when the tornado occurred for the years 2000–2018. Therefore, $\overline{x}$ and $\sigma $ have different values for each individual tornado, both for the ERA-5 and the ERA-Interim datasets.

#### 2.4. Statistical Tools and Procedures

#### 2.4.1. Homogeneity Tests

#### 2.4.2. Conditional Probabilities

#### 2.4.3. Multivariate Analysis via Copulas

## 3. Results

#### 3.1. Relevance of Geographical and Seasonal Features

#### 3.2. Evolution of WS and CAPE before a Tornado Occurrence

#### 3.3. Conditional Probability of Tornado Occurrence

#### 3.4. Multivariate Analysis of Tornado Occurrence

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LLS | wind shear as the difference between the wind speed near the surface (1000 hPa) and at about 1 km (900 hPa) |

DLS | wind shear as the difference between the wind speed near the surface and at about 6 km (500 hPa) |

CAPE | convective available potential energy |

WM | Updraft Maximum Vertical Velocity |

## Appendix A. The Frank Family of Copulas

## Appendix B. ERA-Interim Figures

**Figure A1.**Same as Figure 2 but considering the ERA-Interim dataset.

**Figure A2.**Same as Figure 3 but considering the ERA-Interim dataset.

**Figure A3.**Same as Figure 4 but considering the ERA-Interim dataset.

**Figure A4.**Same as Figure 5 but considering the ERA-Interim dataset.

**Figure A5.**Same as Figure 6 but considering the ERA-Interim dataset.

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**Figure 1.**Locations of EF2+ tornadoes occurred in Italy during the period 2000–2018. The colors indicate the seasons: spring (green), summer (red), and autumn (blue). Different markers represent three different regions: northern Italy (circles), the Tyrrhenian coast (squares), southern Italy (triangles). The increasing size of the markers denote stronger intensity of tornadoes (from EF2 to EF4)—see the corresponding legend.

**Figure 2.**Time Series of updraft maximum vertical velocity (WM) and wind shear (LLS and DLS) extracted from ERA-5 at the 6 h time step closest to the occurrence of the tornadoes (TS1). The height of the bars indicates the magnitude of the variable, while the color marks the season when tornadoes occurred (see also Figure 1).

**Figure 3.**Boxplots of WM, LLS and DLS values in different sub-regions (

**left**) and seasons (

**right**) considering the ERA-5 dataset. The edges of the box indicate the 25th and 75th percentiles. The whiskers correspond to approximately 99% confidence interval.

**Figure 4.**Same as Figure 3 but referred to Standardized indices. The horizontal black dashed line at zero corresponds to the mean climate condition.

**Figure 5.**Estimates (bars) of the probability of tornado occurrence conditional to the fact that WM (

**top-left**), or LLS (

**top-right**), or DLS (

**bottom-left**) takes on a value in a given bin at TS1 (ERA-5 dataset). The vertical blue lines correspond to 95% bootstrap Confidence Intervals. The red lines represent the outcome of the linear regression, and the legends report the main statistical results. The top labels show the conditional probabilities in each bin (% value). The bottom-right panel summarizes and compares the regression p-values.

**Figure 6.**Isolines of the empirical copulas (black lines) of the pairs (

**WM, LLS**), (

**WM, DLS**), and (

**LLS, DLS**) for the ERA-5 dataset, and of the corresponding fitted copulas (red lines). Also shown are the available data (markers). The bottom-right panel shows a comparison between the three fitted copulas.

**Table 1.**The sub-intervals ${\Delta}_{i}$’s used to partition the ranges spanned by WM, LLS, and DLS, for both the ERA-5 and the ERA-Interim datasets: the units are in m/s. The values in brackets indicate the number of tornadoes recorded in each bin.

ERA-5 | ||||

${\Delta}_{1}$Low | ${\Delta}_{2}$Medium | ${\Delta}_{3}$High | ${\Delta}_{4}$Extreme | |

[15] | [14] | [14] | [14] | |

WM | $(-\infty ,28.9]$ | $(28.9,39.9]$ | $(39.9,51.8]$ | $(51.8,+\infty )$ |

LLS | $(-\infty ,5.1]$ | $(5.1,7.9]$ | $(7.9,11.4]$ | $(11.4,+\infty )$ |

DLS | $(-\infty ,15.7]$ | $(15.7,19.5]$ | $(19.5,23.9]$ | $(23.9,+\infty )$ |

ERA-Interim | ||||

${\Delta}_{1}$Low | ${\Delta}_{2}$Medium | ${\Delta}_{3}$High | ${\Delta}_{4}$Extreme | |

[14] | [14] | [13] | [14] | |

WM | $(-\infty ,22.1]$ | $(22.1,36.3]$ | $(36.3,45.8]$ | $(45.8,+\infty )$ |

LLS | $(-\infty ,3.7]$ | $(3.7,6.9]$ | $(6.9,9.1]$ | $(9.1,+\infty )$ |

DLS | $(-\infty ,12.6]$ | $(12.6,16.6]$ | $(16.6,20.7]$ | $(20.7,+\infty )$ |

**Table 2.**Statistical comparison of the samples of WM, LLS, and DLS collected in different regions and seasons: both ERA-5 and ERA-Interim reanalyses are considered. Shown are the p-values (in %) of (i) the KS and AD two-sample homogeneity tests (the Null assumption is that the distributions of the samples are statistically compatible), and (ii) the MW central tendency test (the Null assumption is that it is equally likely that a randomly selected value from one population is less, or greater, than a randomly selected value from a second population). Values in bold are smaller than 1%, while those in italic are smaller than 5%. The Null assumptions could be rejected at $\alpha $-levels larger than the p-values indicated.

Regional Analysis | |||||||

KS | AD | MW | |||||

Region | ERA5 | ERAint. | ERA5 | ERAint. | ERA5 | ERAint. | |

Tyr.-South | 14.30 | 24.59 | 2.10 | 7.03 | 1.70 | 7.74 | |

WM | Tyr.-North | 5.70 | 80.88 | 4.80 | 67.24 | 4.20 | 69.62 |

South-North | 27.70 | 0.99 | 25.50 | 1.73 | 92 | 4.12 | |

Tyr.-South | 99.30 | 43.23 | 98.70 | 56.19 | 89.70 | 39.14 | |

LLS | Tyr.-North | 45.70 | 0.39 | 45.30 | 0.32 | 27.30 | 0.37 |

South-North | 27.70 | 0.06 | 11.80 | 0.01 | 8.60 | 0.01 | |

Tyr.-South | 74.90 | 92.31 | 83.90 | 92.58 | 73.70 | 93.82 | |

DLS | Tyr.-North | 75.20 | 79.59 | 62.10 | 70.94 | 44.50 | 51.39 |

South-North | 26 | 46.31 | 24.50 | 33.06 | 20.70 | 26.64 | |

Seasonal Analysis | |||||||

KS | AD | MW | |||||

Season | ERA5 | ERAint. | ERA5 | ERAint. | ERA5 | ERAint. | |

Spring-Summer | 86.35 | 87.83 | 66.18 | 91.50 | 51.49 | 100 | |

WM | Spring-Autumn | 20.68 | 25.62 | 23.03 | 13.24 | 27.29 | 74.25 |

Summer-Autumn | 31.18 | 13.17 | 11.82 | 5.05 | 10.09 | 27.72 | |

Spring-Summer | 83.08 | 67.56 | 69.92 | 56.88 | 60.74 | 94.47 | |

LLS | Spring-Autumn | 5.69 | 0.06 | 2.20 | 0.02 | 2.38 | 0.02 |

Summer-Autumn | 6.18 | 0.18 | 1.28 | 0.03 | 0.96 | 0.02 | |

Spring-Summer | 93.18 | 67.56 | 91.55 | 70.01 | 75.81 | 91.71 | |

DLS | Spring-Autumn | 63.24 | 96.53 | 61.28 | 85.21 | 100 | 90.49 |

Summer-Autumn | 99.31 | 74.32 | 98.42 | 70.73 | 86.11 | 70.76 |

**Table 3.**Estimated p-values of the KS and AD two-sample homogeneity tests considering the distributions of the variables WM, LLS, and DLS (both actual values and standardized indices) at time-steps TS1 and TS4, for both the ERA-Interim and the ERA-5 datasets.

Actual Values | ||||

ERA-Interim | ERA-5 | |||

KS | AD | KS | AD | |

WM | 0.0006% | 6.1 × 10^{−6}% | 0.019% | 0.00057% |

LLS | 0.18% | 0.82% | 3.8% | 2.6% |

DLS | 3.9% | 4.6% | 6.4% | 1.1% |

Standardised indices | ||||

ERA-Interim | ERA-5 | |||

KS | AD | KS | AD | |

WM | 0.0082% | 0.00057% | 0.16% | 0.053% |

LLS | 1.3% | 0.12% | 0.016% | 0.019% |

DLS | 3.9% | 2.4% | 0.65% | 0.87% |

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**MDPI and ACS Style**

Ingrosso, R.; Lionello, P.; Miglietta, M.M.; Salvadori, G.
A Statistical Investigation of Mesoscale Precursors of Significant Tornadoes: The Italian Case Study. *Atmosphere* **2020**, *11*, 301.
https://doi.org/10.3390/atmos11030301

**AMA Style**

Ingrosso R, Lionello P, Miglietta MM, Salvadori G.
A Statistical Investigation of Mesoscale Precursors of Significant Tornadoes: The Italian Case Study. *Atmosphere*. 2020; 11(3):301.
https://doi.org/10.3390/atmos11030301

**Chicago/Turabian Style**

Ingrosso, Roberto, Piero Lionello, Mario Marcello Miglietta, and Gianfausto Salvadori.
2020. "A Statistical Investigation of Mesoscale Precursors of Significant Tornadoes: The Italian Case Study" *Atmosphere* 11, no. 3: 301.
https://doi.org/10.3390/atmos11030301