This thesis primarily concerns kinematic dynamo action by the 1:1:1 ABC flow, in the highly conducting limit of large magnetic Reynolds number Rm. The flow possesses 24 symmetries, with a symmetry group isomorphic to the group O24 of orientation-preserving transformations of a cube. These symmetries are exploited to break up the linear eigenvalue problem into five distinct symmetry classes, which we label I-V. The thesis discusses how to reduce the scale of the numerical problem to a subset of Fourier modes for a magnetic field in each class, which then may be solved independently to obtain distinct branches of eigenvalues and magnetic field eigenfunctions. Two numerical methods are employed: the first is to time step a magnetic field in a given symmetry class and obtain the growth rate and frequency by measuring the magnetic energy as a function of time. The second method involves a more direct determination of the eigenvalue using the eigenvalue solver ARPACK for sparse matrix systems, which employs an implicitly restarted Arnoldi method. The two methods are checked against each other, and compared for efficiency and reliability. Eigenvalue branches for each symmetry class are obtained for magnetic Reynolds numbers Rm up to 10^4 together with spectra and magnetic field visualisations. A sequence of branches emerges as Rm increases and the magnetic field structures in the different branches are discussed and compared. All symmetry classes are found to contain a dynamo, though dynamo effectiveness varies greatly between classes, suggesting that the symmetries play an important role in the field amplification mechanisms. A closely related problem, that of linear hydrodynamic stability, is also explored in the limit of large Reynolds number Re. As the same symmetry considerations apply, the five symmetry classes of the linear instability can be resolved independently, reducing the size of the problem and allowing exploration of the effects of the symmetries on instability growth rate. Results and visualisations are obtained for all five classes for Re up to 10^3, with comparisons drawn between the structures seen in each class and with those found in the analogous magnetic problem. For increasing Re, multiple mode crossings are observed within each class, with remarkably similar growth rates seen in all classes at Re=10^3, highlighting a lack of dependence on the symmetries of the instability, in contrast with the magnetic problem. This thesis also investigates the problem of large-scale magnetic fields in the 1:1:1 ABC flow through the introduction of Bloch waves that modify the periodicity of the magnetic field relative to the flow. Results are found for a field with increased periodicity in a single direction for Rm up to 10^3; it is established that the optimal scale for dynamo action varies as Rm increases, settling on a consistent scale for large Rm. The emerging field structures are studied and linked with those of the original dynamo problem. On contrasting this method with a previous study in which the flow is instead rescaled, it is shown that the use of Bloch waves drastically increases the range of possible scales, whilst cutting required computing time. Through a multiple-scale analysis, the contribution from the alpha-effect is calculated for the 1:1:1 ABC flow and is seen in growth rates for Rm << 1.

La dynamique des plasmas de fusion par confinement magnétique dans la configuration Reversed Field Pinch (RFP) est ´étudiée en utilisant la description magnétohydrodynamique (MHD) incompressible. Une méthode pseudo-spectrale et une technique de pénalisation en volume sont utilisées pour résoudre le système d’équations dans un cylindre. Les simulations numériques montrent que la pression joue un rôle important dans la dynamique des RFP et ne peut pas être négligée. Ainsi, ß n’est plus le paramètre principal pour décrire la dynamique des RFPs mais plutôt ß’ ∇, un nouveau paramètre qui équivaut le rapport du module de gradient de pression et le module de la force de Lorentz. A un autre niveau, l’effet du changement de la section poloïdale du RFP sur la dynamique est étudié. Les simulations des écoulements RFP ayant le même nombre de Lundquist et des sections différentes (circulaire et elliptique), montrent une grande différence dans les spectres et la diffusion turbulente radiale. Finalement, les écoulements RFP sont utilisés pour étudier l’effet dynamo. Les résultats obtenus montrent que les écoulements RFP sont capables d’amplifier un champ magnétique passif qui aura une tendance à être plus non-linéaire que le champ magnétique du RFP dans les régimes turbulents. / The dynamics of magnetic fusion plasmas in the Reversed Field Pinch (RFP) configuration are studied using an incompressible magnetohydrodynamics (MHD) description. A pseudospectral method combined with a volume penalization method are used to resolve the governing equations in a straight cylinder. Numerical simulations show that the pressure effects on the RFP dynamics cannot be neglected, and thus the _ parameter is not adequate to characterize the importance of pressure in the dynamics. A new parameter, _0r , which is the ratio of the pressure gradient’s magnitude to the Lorentz force’s magnitude, is proposed to be the proper parameter to describe the RFP dynamics. Another investigated influence on the RFP dynamics is the shaping of the poloidal cross-section. Simulations of flows with the same Lundquist number and different cross-sections (circular and elliptic) show a clear change in the spectral behaviour, as well as in the radial turbulent diffusion. Finally, the RFP flows are used to study the dynamo effect. Numerical results show that RFP flows are capable of amplifying a seed magnetic field, which will have tendency to be more nonlinear than the RFP magnetic field in the turbulent regime.