Abstract
In Chap. 6 we have considered linear stability of parity-invariant MHD steady states to large-scale perturbations. Equations, governing stability modes, were obtained from the equations of magnetohydrodynamics by neglecting the terms, quadratic in perturbations. This procedure is legitimate, when development of perturbations is inspected at an initial stage and their amplitudes are small; when the amplitudes are increasing, the accuracy of description by the linearised equations at large time intervals, evidently, deteriorates. It is therefore desirable to derive a system of nonlinear mean-field equations for the perturbations averaged over small scales, in which the large-scale dynamics of the mean perturbations is uncoupled from the evolution of their short-scale component, and which might be used for the study of the evolution of the perturbations at subsequent stages, when nonlinearity sets in.
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Notes
- 1.
The condition of global boundedness is convenient, because the mean of a derivative of a globally bounded field in any fast variable vanishes, for instance
$$ \begin{aligned} \left\langle\!\!\!\left\langle\frac{{\partial{{\mathbf{f}}}}}{\partial x_1}\right\rangle\!\!\!\right\rangle &=\lim_{\tilde{t}\to\infty}\lim_{\ell\to\infty}\frac{{1}}{\tilde{t}\ell^3}\int\limits_0^{\tilde{t}} \int\limits_{-\ell/2}^{\ell/2}\int\limits_{-\ell/2}^{\ell/2}\int\limits_{-\ell/2}^{\ell/2} \frac{{\partial{{\mathbf{f}}}}}{\partial x_1} \hbox{d}{{\mathbf{x}}} \,\hbox{d}t \\ &=\lim_{\tilde{t}\to\infty}\lim_{\ell\to\infty} \frac{{1}}{\tilde{t}\ell^2}\int\limits_0^{\tilde{t}}\int\limits_{-\ell/2}^{\ell/2}\int\limits_{-\ell/2}^{\ell/2} \frac{{{\mathbf{f}}}(\ell/2,x_2,x_3,t)-{{\mathbf{f}}}(-\ell/2,x_2,x_3,t)}{{\ell}} \hbox{d} x_2 \hbox{d}x_3 \hbox{d}t=0. \\ \end{aligned} $$ - 2.
Clearly, not every family of vector fields depending on \(\varepsilon\) can be expressed as power series (7.7). However, this constraint is not too restrictive, because it is natural to apply our stability analysis for the study of the weakly nonlinear evolution of large-scale MHD perturbation modes and their superpositions, and such modes admit asymptotic expansions in power series (7.7), see Chap. 6.
- 3.
Here is a rigorous proof: Denote \(\mu=\inf|p_1(f_1/f_2)-p_2|\), where the infimum is over all integer numbers \(p_1\) and \(p_2\). Evidently, \(0\le\mu<1/2\). Assuming \(\mu>0\), choose integer \(p_1>0\) and \(p_2\ge0\) such that \(\alpha\equiv|p_1(f_1/f_2)-p_2|\) satisfies \(0\le\alpha-\mu\ll\mu\). Denote by \(K\) the integer part of \(\alpha^{-1}\). Evidently, at least one of the two positive numbers: \(1-|Kp_1(f_1/f_2)-Kp_2|\) and \(|(K+1)p_1(f_1/f_2)-(K+1)p_2|-1\) does not exceed \(\alpha/2<\mu\). This, however, contradicts with the definition of \(\mu\); therefore, \(\mu=0\), as required.
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© 2011 Springer-Verlag Berlin Heidelberg
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Zheligovsky, V. (2011). Weakly Nonlinear Stability of MHD Regimes to Large-Scale Perturbations. 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_7
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DOI: https://doi.org/10.1007/978-3-642-18170-2_7
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