Abstract
In this chapter we consider, following [330], weakly nonlinear stability to large-scale perturbations of convective hydromagnetic (CHM) regimes in a horizontal layer of electrically conducting fluid. (A study of the preceding linear stage of their evolution
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Notes
- 1.
This condition is also sufficient for space-periodic regimes. If space periodicity is not imposed, the energy spectrum of the vector field to which the inverse curl is applied must fall off sufficiently fast near zero. Henceforth, the regime \({\mathbf{V}},{\mathbf{H}},\Uptheta\) is assumed to have the property that, in the course of solution of the auxiliary problems, the inverse curl can be applied to any vector field whose vertical component has a zero spatial mean.
- 2.
In this class the bilinear form \(\langle{\mathbf{a}}\cdot{\mathbf{b}}\rangle\) is not a genuine scalar product, because, for instance, \(\langle a^2({\mathbf{x}},t)\rangle=0\) for any smooth field \(a\) with a compact support.
- 3.
\(\beta\) proportional to the Rayleigh number is chosen as the bifurcation parameter here, since the sequence of bifurcations happening on increasing the Rayleigh number was investigated by many authors, see, e.g., [227]. We could consider a bifurcation occurring on variation of any other parameter of the problem, e.g., \(\tau\) proportional to the Taylor number—such bifurcations were examined in [55]. This does not affect the structure of the mean-field and amplitude equations that we derive in this section.
- 4.
See the footnote on p. 171.
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© 2011 Springer-Verlag Berlin Heidelberg
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Zheligovsky, V. (2011). Weakly Nonlinear Stability of Forced Thermal Hydromagnetic Convection. In: Large-Scale Perturbations of Magnetohydrodynamic Regimes. Lecture Notes in Physics, vol 829. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18170-2_8
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DOI: https://doi.org/10.1007/978-3-642-18170-2_8
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