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Scale selection in hydromagnetic dynamos



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Valeria Shumaylova, Valeria 


One of the extraordinary properties of the Sun is the observed range of motion scales from the convection granules to the cyclic variation of magnetic activity. The Sun’s magnetic field exhibits coherence in space and time on much larger scales than the turbulent convection that ultimately powers the dynamo. Motivated by the scale separation considerations, in this thesis we study the parametric scale selection of dynamo action. Although helioseismology has made a lot of progress in the study of the solar interior, the precise motions of plasma are still unknown. In this work, we assume that the model flow is forced with helical viscous body forces acting on different characteristic scales and weak and strong large-scale shear flows that are believed to be present near the base of the convection zone. In this thesis, we look for numerical evidence of a large-scale magnetic field relative to the characteristic scale of the model flow. The investigation is based on the simulations of incompressible MHD equations in elongated triply-periodic domains. To commence the investigation, a linear stability analysis of the coarsening instability in a one-dimensional periodic system is performed to study the stability threshold in the mean-field limit that assumes large scale separation in the system. The simulations are used to discriminate between different forms of the mean-field α -effect and domain aspect ratio. The notion of scale selection refers to methods for estimating characteristic scales. We define the dynamo scale through the characteristic scales of the underlying model flow, forcing and the realised magnetic field. The aspect ratio of the elongated domains plays a crucial role in all considered cases. In Part II, we examine the dynamo generated by the imposed model flows. The transition from large-scale dynamo at the onset to small-scale dynamo as we increase Rm is smooth and takes place in two stages: a fast transition into a predominantly small-scale magnetic energy state and a slower transition into even smaller scales. The long wavelength perturbation imposed on the ABC flow in the modulated case is not preserved in the eigenmodes of the magnetic field. In the presence of the linear (semi-linear shearing-box approximation) and the sinusoidal shearing motions, the field again undergoes a smooth transition at the slow non-sheared rate, which is associated with the balance of the advection and diffusion terms in the induction equation. Part III considers the nonlinear extension of the analysis in Part II, where the incompressible cellular and sheared flows interact with the exponentially growing magnetic field via the Lorentz force in the dynamical regime. Both sheared and non-sheared helical cellular flows become unstable to large-scale perturbations even in the limit of high viscosity. Due to the helical properties of the imposed forcing, the inverse cascade of helicity leads to energy accumulation in the largest scales of the domain, albeit the characteristic lengthscale exhibits the transitional nature at a highly reduced rate in the mean-field limit. As Rm is increased, the transition resembles that of the kinematic regime. The unique properties of the anisotropic shear reduce the componentality of the system, which in turn is able to half the rate of transition from the large-scale dynamo at the onset to a small-scale one.





Proctor, Michael R. E.


Dynamo theory, MHD, ABC flow, Mean-field theory


Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge
Isaac Newton Studentship King's College Hardship Fund