Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary <i>αω</i>-Dynamo Model
We study some fractal properties of the hereditary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mi>ω</mi></mrow></semantics></math></inline-formul...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/6/328 |
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| author | Gleb Vodinchar Lyubov Feshchenko |
| author_facet | Gleb Vodinchar Lyubov Feshchenko |
| author_sort | Gleb Vodinchar |
| collection | DOAJ |
| description | We study some fractal properties of the hereditary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mi>ω</mi></mrow></semantics></math></inline-formula>-dynamo model in the two-mode approximation. The phase variables of the model describe the temporal dynamics of the toroidal and poloidal components of the magnetic field. The hereditary operator of the quenching the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-effect by field helicity in numerical simulation is determined using the Riemann–Liouville fractional differentiation operator. The model also includes a stochastic term. The structure of this term corresponds to the effect of coherent structures from small-scale magnetic field and velocity modes. A difference scheme and a program code for numerical simulation have been developed and verified. A series of computational experiments with the model has been carried out. The Hausdorff dimension of the polarity scale in the model and the distribution of polarity intervals are calculated. It is shown that the Hausdorff dimension of the polarity scale is less than 1, i.e., this scale is a fractal. The numerical value of the dimension for some values of the control parameters is 0.87, which is consistent with the dimension of the real geomagnetic polarity scale. The distribution histogram of polarity intervals in the model has a pronounced power-law tail, which also agrees with the properties of real polarity scales. |
| first_indexed | 2024-03-09T23:45:52Z |
| format | Article |
| id | doaj.art-803f63bf53724ff186a5ea19e72d3689 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T23:45:52Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-803f63bf53724ff186a5ea19e72d36892023-11-23T16:42:52ZengMDPI AGFractal and Fractional2504-31102022-06-016632810.3390/fractalfract6060328Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary <i>αω</i>-Dynamo ModelGleb Vodinchar 0Lyubov Feshchenko1Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 7, Mirnaya Str., Paratunka, 684034 Kamchatka, RussiaInstitute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 7, Mirnaya Str., Paratunka, 684034 Kamchatka, RussiaWe study some fractal properties of the hereditary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mi>ω</mi></mrow></semantics></math></inline-formula>-dynamo model in the two-mode approximation. The phase variables of the model describe the temporal dynamics of the toroidal and poloidal components of the magnetic field. The hereditary operator of the quenching the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-effect by field helicity in numerical simulation is determined using the Riemann–Liouville fractional differentiation operator. The model also includes a stochastic term. The structure of this term corresponds to the effect of coherent structures from small-scale magnetic field and velocity modes. A difference scheme and a program code for numerical simulation have been developed and verified. A series of computational experiments with the model has been carried out. The Hausdorff dimension of the polarity scale in the model and the distribution of polarity intervals are calculated. It is shown that the Hausdorff dimension of the polarity scale is less than 1, i.e., this scale is a fractal. The numerical value of the dimension for some values of the control parameters is 0.87, which is consistent with the dimension of the real geomagnetic polarity scale. The distribution histogram of polarity intervals in the model has a pronounced power-law tail, which also agrees with the properties of real polarity scales.https://www.mdpi.com/2504-3110/6/6/328hereditary systemsfractional dynamicsαω-dynamogeodynamogeomagnetic polarity scalefractal time series |
| spellingShingle | Gleb Vodinchar Lyubov Feshchenko Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary <i>αω</i>-Dynamo Model Fractal and Fractional hereditary systems fractional dynamics αω-dynamo geodynamo geomagnetic polarity scale fractal time series |
| title | Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary <i>αω</i>-Dynamo Model |
| title_full | Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary <i>αω</i>-Dynamo Model |
| title_fullStr | Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary <i>αω</i>-Dynamo Model |
| title_full_unstemmed | Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary <i>αω</i>-Dynamo Model |
| title_short | Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary <i>αω</i>-Dynamo Model |
| title_sort | fractal properties of the magnetic polarity scale in the stochastic hereditary i αω i dynamo model |
| topic | hereditary systems fractional dynamics αω-dynamo geodynamo geomagnetic polarity scale fractal time series |
| url | https://www.mdpi.com/2504-3110/6/6/328 |
| work_keys_str_mv | AT glebvodinchar fractalpropertiesofthemagneticpolarityscaleinthestochastichereditaryiaōidynamomodel AT lyubovfeshchenko fractalpropertiesofthemagneticpolarityscaleinthestochastichereditaryiaōidynamomodel |