A Century of Turbulent Cascades and the Emergence of Multifractal Operators
Abstract A century of cascades and three decades of multifractals have built up a truly interdisciplinary framework that has enabled a new approach and understanding of nonlinear phenomena, in particular, in geophysics. Nevertheless, there seems to be a profound gap between the potentials of multifr...
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| Format: | Article |
| Language: | English |
| Published: |
American Geophysical Union (AGU)
2020-03-01
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| Series: | Earth and Space Science |
| Subjects: | |
| Online Access: | https://doi.org/10.1029/2019EA000608 |
| _version_ | 1811312785475239936 |
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| author | Daniel Schertzer Ioulia Tchiguirinskaia |
| author_facet | Daniel Schertzer Ioulia Tchiguirinskaia |
| author_sort | Daniel Schertzer |
| collection | DOAJ |
| description | Abstract A century of cascades and three decades of multifractals have built up a truly interdisciplinary framework that has enabled a new approach and understanding of nonlinear phenomena, in particular, in geophysics. Nevertheless, there seems to be a profound gap between the potentials of multifractals and their actual use. For instance, it seems ironic that multifractals have been mostly restricted to scalar‐valued fields, whereas cascades were first invoked for the wind velocity. We argue that this requires to proceed to new developments of the multifractal formalism and to the emergence of multifractal operators. This paper therefore aims to first simplify the introduction to the most recent developments based on the analysis and generation of multifractal fields with the help of the group property of the responses of a nonlinear system to a scale change. The generators of the multifractal operators are introduced with the help of symmetries as simple and basic as orthogonal rotations and mirror symmetries. This leads in a rather straightforward manner to the large class of Gauss–Clifford and Lévy–Clifford generators that combine a number of seductive properties, including universal statistical and robust algebraic properties. At the same time, we obtain new results on the entanglement of spherical and hyperbolic geometries, as well as on the existence of finite statistics of these cascades. |
| first_indexed | 2024-04-13T10:42:46Z |
| format | Article |
| id | doaj.art-932f9920843640378f76e5d96969be6a |
| institution | Directory Open Access Journal |
| issn | 2333-5084 |
| language | English |
| last_indexed | 2024-04-13T10:42:46Z |
| publishDate | 2020-03-01 |
| publisher | American Geophysical Union (AGU) |
| record_format | Article |
| series | Earth and Space Science |
| spelling | doaj.art-932f9920843640378f76e5d96969be6a2022-12-22T02:49:52ZengAmerican Geophysical Union (AGU)Earth and Space Science2333-50842020-03-0173n/an/a10.1029/2019EA000608A Century of Turbulent Cascades and the Emergence of Multifractal OperatorsDaniel Schertzer0Ioulia Tchiguirinskaia1Hydrology Meteorology and Complexity Ecole des Ponts ParisTech Marne‐la‐Vallée FranceHydrology Meteorology and Complexity Ecole des Ponts ParisTech Marne‐la‐Vallée FranceAbstract A century of cascades and three decades of multifractals have built up a truly interdisciplinary framework that has enabled a new approach and understanding of nonlinear phenomena, in particular, in geophysics. Nevertheless, there seems to be a profound gap between the potentials of multifractals and their actual use. For instance, it seems ironic that multifractals have been mostly restricted to scalar‐valued fields, whereas cascades were first invoked for the wind velocity. We argue that this requires to proceed to new developments of the multifractal formalism and to the emergence of multifractal operators. This paper therefore aims to first simplify the introduction to the most recent developments based on the analysis and generation of multifractal fields with the help of the group property of the responses of a nonlinear system to a scale change. The generators of the multifractal operators are introduced with the help of symmetries as simple and basic as orthogonal rotations and mirror symmetries. This leads in a rather straightforward manner to the large class of Gauss–Clifford and Lévy–Clifford generators that combine a number of seductive properties, including universal statistical and robust algebraic properties. At the same time, we obtain new results on the entanglement of spherical and hyperbolic geometries, as well as on the existence of finite statistics of these cascades.https://doi.org/10.1029/2019EA000608turbulence cascademultifractatalsintermittencyscaling vector fieldsLevy stable vectorsClifford algebra |
| spellingShingle | Daniel Schertzer Ioulia Tchiguirinskaia A Century of Turbulent Cascades and the Emergence of Multifractal Operators Earth and Space Science turbulence cascade multifractatals intermittency scaling vector fields Levy stable vectors Clifford algebra |
| title | A Century of Turbulent Cascades and the Emergence of Multifractal Operators |
| title_full | A Century of Turbulent Cascades and the Emergence of Multifractal Operators |
| title_fullStr | A Century of Turbulent Cascades and the Emergence of Multifractal Operators |
| title_full_unstemmed | A Century of Turbulent Cascades and the Emergence of Multifractal Operators |
| title_short | A Century of Turbulent Cascades and the Emergence of Multifractal Operators |
| title_sort | century of turbulent cascades and the emergence of multifractal operators |
| topic | turbulence cascade multifractatals intermittency scaling vector fields Levy stable vectors Clifford algebra |
| url | https://doi.org/10.1029/2019EA000608 |
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