Possible implications of self-similarity for tornadogenesis and maintenance

Pavel Bělík; Brittany Dahl; Douglas Dokken; Corey K. Potvin; Kurt Scholz; Mikhail Shvartsman; Pavel Bělík; Brittany Dahl; Douglas Dokken; Corey K. Potvin; Kurt Scholz; Mikhail Shvartsman

doi:10.3934/Math.2018.3.365

AIMS Mathematics

2018, Volume 3, Issue 3: 365-390. doi: 10.3934/Math.2018.3.365

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Research article Topical Sections

Possible implications of self-similarity for tornadogenesis and maintenance

1.
Department of Mathematics, Statistics, and Computer Science, Augsburg University, Minneapolis, MN 55454, USA
2.
Cooperative Institute for Marine and Atmospheric Studies, University of Miami, and NOAA/Atlantic Oceanographic and Meteorological Laboratory/Hurricane Research Division, Miami, FL, USA
3.
Department of Mathematics, University of St. Thomas, St. Paul, MN 55105, USA
4.
Cooperative Institute for Mesoscale Meteorological Studies, and School of Meteorology, University of Oklahoma, and NOAA/OAR/National Severe Storms Laboratory, Norman, OK, USA

Received: 12 July 2018 Accepted: 28 September 2018 Published: 10 October 2018
MSC : 28A80, 76B47, 76D05, 76F06, 76F10, 76M55, 76U05, 86A10

Self-similarity in tornadic and some non-tornadic supercell flows is studied and power laws relating various quantities in such flows are demonstrated. Magnitudes of the exponents in these power laws are related to the intensity of the corresponding flow and thus the severity of the supercell storm. The features studied in this paper include the vertical vorticity and pseudovorticity, both obtained from radar observations and from numerical simulations, the tangential velocity, and the energy spectrum as a function of the wave number. Power laws for the vertical vorticity, pseudovorticity, and tangential velocity obtained from radar observations studied in the literature are summarized. Further support is given to the existence of a power law for vorticity by the analysis of data obtained from a numerical simulation of a tornadic supercell. A possible explanation for an increase in vorticity in a developing tornado is provided, as well as an argument that tornadoes have approximate fractal cross sections and negative temperatures. A power law that relates the increase of the approximate fractal dimension of the cross section of a negative temperature vortex to its energy content is derived, and the possible applicability of the box-counting method to determine this quantity from suitable radar images is demonstrated.
- tornado,
- tornadogenesis,
- power laws,
- self-similarity,
- fractal,
- fractal dimension,
- vorticity,
- pseudovorticity,
- energy spectrum
Citation: Pavel Bělík, Brittany Dahl, Douglas Dokken, Corey K. Potvin, Kurt Scholz, Mikhail Shvartsman. Possible implications of self-similarity for tornadogenesis and maintenance[J]. AIMS Mathematics, 2018, 3(3): 365-390. doi: 10.3934/Math.2018.3.365

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Abstract

Self-similarity in tornadic and some non-tornadic supercell flows is studied and power laws relating various quantities in such flows are demonstrated. Magnitudes of the exponents in these power laws are related to the intensity of the corresponding flow and thus the severity of the supercell storm. The features studied in this paper include the vertical vorticity and pseudovorticity, both obtained from radar observations and from numerical simulations, the tangential velocity, and the energy spectrum as a function of the wave number. Power laws for the vertical vorticity, pseudovorticity, and tangential velocity obtained from radar observations studied in the literature are summarized. Further support is given to the existence of a power law for vorticity by the analysis of data obtained from a numerical simulation of a tornadic supercell. A possible explanation for an increase in vorticity in a developing tornado is provided, as well as an argument that tornadoes have approximate fractal cross sections and negative temperatures. A power law that relates the increase of the approximate fractal dimension of the cross section of a negative temperature vortex to its energy content is derived, and the possible applicability of the box-counting method to determine this quantity from suitable radar images is demonstrated.

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