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1.
Chen, Long.
Optimization of *Kinematic* Dynamos Using Variational Methods.

Degree: 2018, ETH Zürich

URL: http://hdl.handle.net/20.500.11850/227237

The Earth possesses a magnetic field that is generated by the fluid motion in a conducting outer core. This system that converts kinetic energy into long lasting magnetic energy is called a dynamo. Not only found on the Earth, a dynamo is a fundamental mechanism that also exists in astrophysical bodies, and various research groups have reproduced dynamos with computer simulations and experiments. Despite extensive studies there is no general recipe to guarantee dynamo action. One important question is therefore: how to generate a dynamo most efficiently? In this thesis, we adapt a variational method to search numerically for the most efficient dynamos and the corresponding optimal flow fields. This method covers a large parameter space that in theory represents infinitely many field configurations, something conventional methods cannot achieve.
Our optimization scheme combines existing dynamo models with adjoint modelling and subsequent updates using variational derivatives. We start with a kinematic dynamo model and update iteratively the initial conditions of both a steady flow field and a magnetic field. We use the enstrophy based magnetic Reynolds number (Rm) as an input parameter. For a given Rm, the asymptotic growth of the magnetic energy needs to be non-negative in order to maintain a dynamo. When the asymptotic growth is precisely zero in an optimized model, we identify the corresponding value of Rm as the lower bound for dynamo action, denoted by the minimal critical magnetic Reynolds number Rm_{c,min}. For some non-dynamo configurations the magnetic energy can grow during a transient period but eventually decays. The critical transient magnetic Reynolds number for which the magnetic energy cannot grow in any time window, even a very narrow one, is denoted by Rm_{t}.
Using this method, we study kinematic dynamos in three main categories: unconstrained dynamos in a cube, unconstrained dynamos in a full sphere and dynamos with symmetries in a full sphere. All models are implemented numerically using a spectral Galerkin method. In the cubic model, we study optimized dynamos at Rm_{c,min} with four sets of magnetic boundary conditions: NNT, NTT, NNN and TTT (T denotes superconducting boundary conditions and N denotes pseudo-vacuum boundary conditions on opposite sides of the cube), meanwhile keeping the flow field satisfying impermeable boundary conditions. Numerically swapping the magnetic boundary conditions from T to N leaves the magnetic energy growth nearly unchanged, and if \mathbf{u} is an optimal flow field, then - \mathbf{u} is the new optimum after swapping. For the mixed cases, we can represent the dominant optimal flow field at Rm_{c,min} with three Fourier modes that each describe a 2D flow field.
In the unconstrained spherical models, we impose electrically insulating boundary conditions on the magnetic field while we let the flow field satisfy either no-slip or free-slip boundary conditions. For the no-slip case, we find the optimal flow at Rm_{c,min} is spatially…
*Advisors/Committee Members: Jackson, Andrew, Noir, Jérõme André Roland, Willis, Ashley.*

Subjects/Keywords: Dynamo theory; Variational methods; Kinematic dynamo; Optimization; info:eu-repo/classification/ddc/550; info:eu-repo/classification/ddc/530; Earth sciences; Physics

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APA (6^{th} Edition):

Chen, L. (2018). Optimization of Kinematic Dynamos Using Variational Methods. (Doctoral Dissertation). ETH Zürich. Retrieved from http://hdl.handle.net/20.500.11850/227237

Chicago Manual of Style (16^{th} Edition):

Chen, Long. “Optimization of Kinematic Dynamos Using Variational Methods.” 2018. Doctoral Dissertation, ETH Zürich. Accessed January 25, 2021. http://hdl.handle.net/20.500.11850/227237.

MLA Handbook (7^{th} Edition):

Chen, Long. “Optimization of Kinematic Dynamos Using Variational Methods.” 2018. Web. 25 Jan 2021.

Vancouver:

Chen L. Optimization of Kinematic Dynamos Using Variational Methods. [Internet] [Doctoral dissertation]. ETH Zürich; 2018. [cited 2021 Jan 25]. Available from: http://hdl.handle.net/20.500.11850/227237.

Council of Science Editors:

Chen L. Optimization of Kinematic Dynamos Using Variational Methods. [Doctoral Dissertation]. ETH Zürich; 2018. Available from: http://hdl.handle.net/20.500.11850/227237

2.
Jones, Samuel Edward.
Symmetries in the *kinematic* dynamos and hydrodynamic instabilities of the ABC flows.

Degree: PhD, 2013, University of Exeter

URL: http://hdl.handle.net/10871/14689

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…

Subjects/Keywords: 532; Kinematic dynamo; ABC flow; Symmetries; Fast dynamo; Arnoldi method; Bloch wave

…eigenvalues of the *kinematic* 1:1:1 ABC *dynamo* 156
Comparison of time stepping and Arnoldi solver… …*kinematic* *dynamo* theory
that are most relevant to this thesis.
1.3.1
Antidynamo theorems
Within… …into the effect of large-scale
magnetic fields on *dynamo* growth.
1.3.3
*Kinematic* *dynamo*… …theory
In the previous section, the main assumption for *kinematic* *dynamo* theory was
given by… …amplification.
*Kinematic* (linear) *dynamo* theory is useful for ascertaining whether specific…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Jones, S. E. (2013). Symmetries in the kinematic dynamos and hydrodynamic instabilities of the ABC flows. (Doctoral Dissertation). University of Exeter. Retrieved from http://hdl.handle.net/10871/14689

Chicago Manual of Style (16^{th} Edition):

Jones, Samuel Edward. “Symmetries in the kinematic dynamos and hydrodynamic instabilities of the ABC flows.” 2013. Doctoral Dissertation, University of Exeter. Accessed January 25, 2021. http://hdl.handle.net/10871/14689.

MLA Handbook (7^{th} Edition):

Jones, Samuel Edward. “Symmetries in the kinematic dynamos and hydrodynamic instabilities of the ABC flows.” 2013. Web. 25 Jan 2021.

Vancouver:

Jones SE. Symmetries in the kinematic dynamos and hydrodynamic instabilities of the ABC flows. [Internet] [Doctoral dissertation]. University of Exeter; 2013. [cited 2021 Jan 25]. Available from: http://hdl.handle.net/10871/14689.

Council of Science Editors:

Jones SE. Symmetries in the kinematic dynamos and hydrodynamic instabilities of the ABC flows. [Doctoral Dissertation]. University of Exeter; 2013. Available from: http://hdl.handle.net/10871/14689

3. Chahine, Robert. MHD simulations of the Reversed Field Pinch : Simulations MHD du Reversed Field Pinch.

Degree: Docteur es, Mécanique des fluides, 2017, Lyon

URL: http://www.theses.fr/2017LYSEC056

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.

Subjects/Keywords: Magnétohydrodynamique; Reversed field pinch; Gradient de pression; Effet de la géométrie; Modes toroïdaux; Diffusion turbulente; Dynamo cinématique; Magnetohydrodynamics; Reversed field pinch; Pressure gradient; Shaping effect; Toroidal modes; Turbulent diffusion; Kinematic dynamo

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Chahine, R. (2017). MHD simulations of the Reversed Field Pinch : Simulations MHD du Reversed Field Pinch. (Doctoral Dissertation). Lyon. Retrieved from http://www.theses.fr/2017LYSEC056

Chicago Manual of Style (16^{th} Edition):

Chahine, Robert. “MHD simulations of the Reversed Field Pinch : Simulations MHD du Reversed Field Pinch.” 2017. Doctoral Dissertation, Lyon. Accessed January 25, 2021. http://www.theses.fr/2017LYSEC056.

MLA Handbook (7^{th} Edition):

Chahine, Robert. “MHD simulations of the Reversed Field Pinch : Simulations MHD du Reversed Field Pinch.” 2017. Web. 25 Jan 2021.

Vancouver:

Chahine R. MHD simulations of the Reversed Field Pinch : Simulations MHD du Reversed Field Pinch. [Internet] [Doctoral dissertation]. Lyon; 2017. [cited 2021 Jan 25]. Available from: http://www.theses.fr/2017LYSEC056.

Council of Science Editors:

Chahine R. MHD simulations of the Reversed Field Pinch : Simulations MHD du Reversed Field Pinch. [Doctoral Dissertation]. Lyon; 2017. Available from: http://www.theses.fr/2017LYSEC056