Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries
We consider Bloch eigenmodes in three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states. A Bloch mode is a product of a field of the same periodicity, as the state subjected to perturbation, and a planar...
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Russian Academy of Sciences, The Geophysical Center
2023-07-01
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| Series: | Russian Journal of Earth Sciences |
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| Online Access: | http://doi.org/10.2205/2023ES000834 |
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| author | Chertovskih R Zheligovsky V |
| author_facet | Chertovskih R Zheligovsky V |
| author_sort | Chertovskih R |
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| description | We consider Bloch eigenmodes in three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states. A Bloch mode is a product of a field of the same periodicity, as the state subjected to perturbation, and a planar harmonic wave, eiq·x. The complex exponential cancels out from the equations of the respective eigenvalue problem, and the wave vector q remains in the equations as a numeric parameter. The resultant problem has a significant advantage from the numerical viewpoint: while the Bloch mode involves two independent spatial scales, its growth rate can be computed in the periodicity box of the perturbed state. The three-dimensional space, where q resides, splits into a number of regions, inside which the growth rate is a smooth function of q. In preparation for a numerical study of the dominant (i.e., the largest over q) growth rates, we have derived expressions for the gradient of the growth rate in q and proven that, for parity-invariant flows and MHD steady states or when the respective eigenvalue of the stability operator is real, half-integer q (whose all components are integer or half-integer) are stationary points of the growth rate. In prior works it was established by asymptotic methods that high spatial scale separation (small q) gives rise to the phenomena of the α-effect or, for parity-invariant steady states, of the eddy diffusivity. We review these findings tailoring them to the prospective numerical applications. |
| format | Article |
| id | doaj-art-4109217e4da14bd9b640e539f3b7cf57 |
| institution | DOAJ |
| issn | 1681-1208 |
| language | English |
| publishDate | 2023-07-01 |
| publisher | Russian Academy of Sciences, The Geophysical Center |
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| series | Russian Journal of Earth Sciences |
| spelling | doaj-art-4109217e4da14bd9b640e539f3b7cf572025-08-20T02:46:35ZengRussian Academy of Sciences, The Geophysical CenterRussian Journal of Earth Sciences1681-12082023-07-0123312010.2205/2023ES000834Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminariesChertovskih R0https://orcid.org/0000-0002-5179-4344Zheligovsky V1Institute of Earthquake Prediction Theory and Mathematical Geophysics Russian academy of sciencesInstitute of Earthquake Prediction Theory and Mathematical Geophysics Russian academy of sciencesWe consider Bloch eigenmodes in three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states. A Bloch mode is a product of a field of the same periodicity, as the state subjected to perturbation, and a planar harmonic wave, eiq·x. The complex exponential cancels out from the equations of the respective eigenvalue problem, and the wave vector q remains in the equations as a numeric parameter. The resultant problem has a significant advantage from the numerical viewpoint: while the Bloch mode involves two independent spatial scales, its growth rate can be computed in the periodicity box of the perturbed state. The three-dimensional space, where q resides, splits into a number of regions, inside which the growth rate is a smooth function of q. In preparation for a numerical study of the dominant (i.e., the largest over q) growth rates, we have derived expressions for the gradient of the growth rate in q and proven that, for parity-invariant flows and MHD steady states or when the respective eigenvalue of the stability operator is real, half-integer q (whose all components are integer or half-integer) are stationary points of the growth rate. In prior works it was established by asymptotic methods that high spatial scale separation (small q) gives rise to the phenomena of the α-effect or, for parity-invariant steady states, of the eddy diffusivity. We review these findings tailoring them to the prospective numerical applications.http://doi.org/10.2205/2023ES000834Kinematic dynamo problem hydrodynamic linear stability problem magnetohydrodynamic linear stability problem Bloch mode magnetic α-effect AKA-effect combined magnetohydrodynamic α-effect magnetic eddy diffusivity eddy viscosity scale separation. |
| spellingShingle | Chertovskih R Zheligovsky V Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries Russian Journal of Earth Sciences Kinematic dynamo problem hydrodynamic linear stability problem magnetohydrodynamic linear stability problem Bloch mode magnetic α-effect AKA-effect combined magnetohydrodynamic α-effect magnetic eddy diffusivity eddy viscosity scale separation. |
| title | Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries |
| title_full | Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries |
| title_fullStr | Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries |
| title_full_unstemmed | Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries |
| title_short | Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries |
| title_sort | linear perturbations of the bloch type of space periodic magnetohydrodynamic steady states i mathematical preliminaries |
| topic | Kinematic dynamo problem hydrodynamic linear stability problem magnetohydrodynamic linear stability problem Bloch mode magnetic α-effect AKA-effect combined magnetohydrodynamic α-effect magnetic eddy diffusivity eddy viscosity scale separation. |
| url | http://doi.org/10.2205/2023ES000834 |
| work_keys_str_mv | AT chertovskihr linearperturbationsoftheblochtypeofspaceperiodicmagnetohydrodynamicsteadystatesimathematicalpreliminaries AT zheligovskyv linearperturbationsoftheblochtypeofspaceperiodicmagnetohydrodynamicsteadystatesimathematicalpreliminaries |