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Title Optimization of Kinematic Dynamos Using Variational Methods
Publication Date
Degree Level doctoral
University/Publisher ETH Zürich
Abstract 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…
Subjects/Keywords Dynamo theory; Variational methods; Kinematic dynamo; Optimization; info:eu-repo/classification/ddc/550; info:eu-repo/classification/ddc/530; Earth sciences; Physics
Contributors Jackson, Andrew; Noir, Jérõme André Roland; Willis, Ashley
Language en
Rights info:eu-repo/semantics/openAccess ; http://rightsstatements.org/page/InC-NC/1.0/ ; In Copyright - Non-Commercial Use Permitted
Country of Publication ch
Format application/application/pdf
Record ID handle:20.500.11850/227237
Other Identifiers info:doi:10.3929/ethz-b-000227237
Repository ethz
Date Retrieved
Date Indexed 2020-08-31

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