Mean-Field Magnetohydrodynamics and Dynamo Theory

Krause, F.; Raedler, K. H.

doi:10.1016/c2013-0-03269-0

Cited by 144 publications

(4 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We outline here the main ideas of the derivation of the combined MHD α-effect and diffusivity tensors in the multiscale linear stability theory for MHD steady states residing in the entire space [Zheligovsky, 2003] and tune the results for the purposes of the present investigation. Let us stress, that we inspect exclusively the results that are obtained from the first principles by asymptotic methods for systems, where a significant scale separation is present; we are not interested here in the rich variety of results of the mean-field electrodynamics relying on additional assumptions, such as the first-order smoothing approximation (also referred to as the second-order correlation approximation), see, e.g., [Krause and Rädler, 1980] and the reviews [Brandenburg and Subramanian, 2005;Brandenburg et al, 2012].…”

Section: The Formalism Of the Multiscale Stability Theorymentioning

confidence: 99%

“…It is built on the seminal idea of E. Parker, who suggested [Parker, 1955] that the interaction of small-scale fluctuations of the flow ("cyclonic events") and of magnetic field may give rise to an electromotive force that has a component parallel to the mean magnetic field, and it can amplify the mean field. A systematic analysis of this idea under various simplifying assumptions was carried out by German scientists [Krause and Rädler, 1980;Steenbeck et al, 1971], who developed the theory of the mean-field magnetohydrodynamics and coined the term α-effect (see [Rädler, 2007] for an account of the history of the subject).…”

Section: Introductionmentioning

confidence: 99%

“…In principle, any averaging procedure satisfying the Reynolds rules (see, e.g., [Krause and Rädler, 1980]) can be employed to define mean fields. Historically, the magnetic α-effect in a weakly non-axisymmetric flow was the first example of the α-effect derived rigorously by asymptotic methods from the first principles [Braginskii, 1964a,b].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries

Chertovskih¹,

Zheligovsky

2023

Russian Journal of Earth Sciences

View full text Add to dashboard Cite

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, exp(iqx). 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.

show abstract

Section: The Formalism Of the Multiscale Stability Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries

Chertovskih¹,

Zheligovsky

2023

Russian Journal of Earth Sciences

View full text Add to dashboard Cite

show abstract

“…When the MHD steady state experiencing a perturbation is parity-invariant (i.e., satisfies V(x) = −V(−x) and B(x) = −B(−x)), then the α-effect disappears (the αeffect tensor vanishes). When λ 0 = λ 1 = 0, the predominant instability mechanism based on scale separation is eddy diffusivity (in particular, eddy viscosity in the context of the hydrodynamic stability problem), aka the β-effect in the speak of the mean-field magnetohydrodynamics (see [Krause and Radler, 1980]). The mean stability eigenfield is then an eigenfunction of the eddy diffusivity operator (see [Chertovskih and Zheligovsky, 2023;Zheligovsky, 2011]).…”

Section: Introductionmentioning

confidence: 99%

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. II. Numerical results

Chertovskih,

Zheligovsky

2023

Russian Journal of Earth Sciences

View full text Add to dashboard Cite

We consider Bloch eigenmodes of three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic ﬂows and MHD states comprised of randomly generated Fourier coeﬃcients and having energy spectra of three types: exponentially decaying, Kolmogorov with a cut oﬀ, or involving a small number of harmonics (“big eddies”). A Bloch mode is a product of a ﬁeld of the same periodicity as the perturbed state and a planar harmonic wave, exp(iq · x). Such a mode is characterized by the ratio of spatial scales which, for simplicity, we identify with the length |q| < 1 of the Bloch wave vector q. Computations have revealed that the Bloch modes, whose growth rates are maximum over q, feature the scale ratio that decreases on increasing the nondimensionalized molecular diﬀusivity and/or viscosity from 0.03 to 0.3, and the scale separation is high (i.e., |q| is small) only for large molecular diﬀusivities. Largely this conclusion holds for all the three stability problems and all the three energy spectra types under consideration. Thus, in a natural MHD system not aﬀected by strong diﬀusion, a given scale range gives rise to perturbations involving only moderately larger spatial scales (i.e., |q| only moderately small), and the MHD evolution consists of a cascade of processes, each generating a slightly larger spatial scale; ﬂows or magnetic ﬁelds characterized by a high scale separation are not produced. This cascade is unlikely to be amenable to a linear description. Consequently, our results question the allegedly high role of the α-eﬀect and eddy diﬀusivity that are based on spatial scale separation, as the primary instability or magnetic ﬁeld generating mechanisms in astrophysical applications. The Braginskii magnetic α-eﬀect in a weakly non-axisymmetric ﬂow, often used for explanation of the solar and geodynamo, is advantageous not being upset by a similar deﬁciency.

show abstract

How MHD Transformed the Theory of Geomagnetism

Roberts

2007

Fluid Mechanics and Its Applications

View full text Add to dashboard Cite

Mean-Field Magnetohydrodynamics and Dynamo Theory

Cited by 144 publications

References 0 publications

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. II. Numerical results

How MHD Transformed the Theory of Geomagnetism

Contact Info

Product

Resources

About