Optimization of kinematic dynamos using variational methods
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| 100 | 1 | |a Chen |D Long |d 1988- |0 (DE-588)1166890090 |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Optimization of kinematic dynamos using variational methods |c presented by Long Chen |
| 264 | 1 | |a Zurich |c 2018 | |
| 300 | |a ix, 178 Seiten |b Illustrationen |c 30 cm | ||
| 502 | |b Dissertation |o No. 24548 |c ETH Zurich, |d 2018 | ||
| 504 | |a Literaturverzeichnis | ||
| 516 | |a application/pdf | ||
| 520 | 3 | |a 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 localized near the centre of the sphere. The dominant optimal flow is well-represented by the first three spherical harmonic degrees $l\leq 3$ and contains only even spherical harmonic order $m$. We also find that the corresponding optimal flow field at $Rm_t$ is equatorially symmetric ($E^S$). For the free-slip case, we get similar results as in the no-slip case, which suggests that the boundary condition of the flow field does not play a significant role in this kinematic model. Previous studies have used symmetry as a guide to categorize dynamo solutions with a simple spectral representation ($\mathcal{O}(10)$ spectral coefficients). We extend this approach here using our large-scale optimizations. We study in total five different set-ups: (1) dynamos generated by axisymmetric flows, (2) dynamos with an equatorially anti-symmetric ($E^A$) magnetic field (the dipole family) generated by $E^S$ flows, (3) dynamos with an $E^S$ magnetic field (the quadrupole family) generated by $E^S$ flows, (4) dynamos generated by axisymmetric $E^S$ flows, (5) dynamos generated by axisymmetric $E^A$ flows. All models in this part satisfy electrically insulating and no-slip boundary conditions. Ranking them by the associated value of $Rm_{c,min}$ from low to high, we find the order (2), (3), (1), (4) and (5). Both set-ups (2) and (3) have the same $Rm_t$ as in the unconstrained case. The most unstable magnetic eigenmode at $Rm_t$ in set-up (4) is $m=0$, but in set-up (5) it is $m=1$. | |
| 546 | |a Englischer Text mit englischer und französischer Zusammenfassung | ||
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