cam a22 4 4500 528992872 CHVBK 20201202100920.0 180926s2018 sz a m 00 0deng d urn:nbn:ch:bel-1675781 urn (SNL)991004718529703976 (SNL)991018080957403976 (NEBIS)011277448 (Sz)1952005-41snl_51 (Sz)vtls001952005 (Sz)001952005 (Sz)991018080957403976 Sz ger rda eng eng fre 530 23sdnb 550 530 23sdnb Chen Long 1988- (DE-588)1166890090 Verfasser aut Optimization of kinematic dynamos using variational methods presented by Long Chen Zurich 2018 ix, 178 Seiten Illustrationen 30 cm Dissertation No. 24548 ETH Zurich, 2018 Literaturverzeichnis application/pdf 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 $m=1$. Englischer Text mit englischer und französischer Zusammenfassung COMPUTATIONAL FLUID DYNAMICS (FLUIDMECHANIK) ger (ETHUDK)000063441 ethudk ERDMAGNETISMUS + GEOMAGNETISMUS (GEOPHYSIK) ger (ETHUDK)000019898 ethudk GALERKIN-VERFAHREN (NUMERISCHE MATHEMATIK) ger (ETHUDK)000042981 ethudk MAGNETOHYDRODYNAMIK ger (ETHUDK)000015237 ethudk MODELLRECHNUNG IN DER GEOPHYSIK ger (ETHUDK)000043373 ethudk REYNOLDS-ZAHL (FLUIDDYNAMIK) ger (ETHUDK)000014579 ethudk Hochschulschrift (DE-588)4113937-9 gnd-content Hochschulschrift gnd-content sb 2018/21 530 zh snl-sb xehelv xediss 2020 550 530 snl ehelv u ERDMAGNETISMUS + GEOMAGNETISMUS (GEOPHYSIK) ger 550.38 nebis E1 u MAGNETOHYDRODYNAMIK ger 537.84 nebis E1 u REYNOLDS-ZAHL (FLUIDDYNAMIK) ger 533.6.011.12 nebis E1 u GALERKIN-VERFAHREN (NUMERISCHE MATHEMATIK) ger 519.65,2 nebis E1 u MODELLRECHNUNG IN DER GEOPHYSIK ger 550.3%519.876.5 nebis E1 u COMPUTATIONAL FLUID DYNAMICS (FLUIDMECHANIK) ger 532.4 nebis E1 u MAGNÉTISME TERRESTRE + GÉOMAGNÉTISME (GÉOPHYSIQUE) fre 550.38 nebis E1 u TERRESTRIAL MAGNETISM + GEOMAGNETISM (GEOPHYSICS) eng 550.38 nebis E1 u MAGNÉTOHYDRODYNAMIQUE fre 537.84 nebis E1 u MAGNETOHYDRODYNAMICS eng 537.84 nebis E1 u REYNOLDS/NOMBRE DE (DYNAMIQUE DES FLUIDES) fre 533.6.011.12 nebis E1 u REYNOLDS NUMBER (FLUID DYNAMICS) eng 533.6.011.12 nebis E1 u MÉTHODE DE GALERKIN (ANALYSE NUMÉRIQUE) fre 519.65,2 nebis E1 u GALERKIN METHOD (NUMERICAL MATHEMATICS) eng 519.65,2 nebis E1 u MODÉLISATION MATHÉMATIQUE EN GÉOPHYSIQUE fre 550.3%519.876.5 nebis E1 u MATHEMATICAL MODELLING IN GEOPHYSICS eng 550.3%519.876.5 nebis E1 u COMPUTATIONAL FLUID DYNAMICS (FLUID MECHANICS) eng 532.4 nebis E1 u DYNAMIQUE NUMÉRIQUE DES FLUIDES (MÉCANIQUE DES FLUIDES) fre 532.4 nebis E1 Eidgenössische Technische Hochschule Zürich (DE-588)2023928-2 Grad-verleihende Institution dgg SNL NB001 NB999 - chb99 - https://opac.admin.ch/toc/toc1902037326.pdf Inhaltsverzeichnis BK020300 XK020000 XK020000 071 E01-20180917 143 E01-20180917 SNL NB001 NB001 Nbq 117635 NB10010 NEBIS E04 E04 E04BI Bg 1054 NEBIS E16 E16 RH Diss ETH 24548 SNL 100 1- Chen Long 1988- (DE-588)1166890090 Verfasser aut SNL 856 41 https://opac.admin.ch/toc/toc1902037326.pdf Inhaltsverzeichnis SNL 710 2- Eidgenössische Technische Hochschule Zürich (DE-588)2023928-2 Grad-verleihende Institution dgg SNL 100 1- Chen Long Verfasser aut SNL 856 40 http://hdl.handle.net/20.500.11850/227237 SNL 856 4- https://nbn-resolving.org/urn:nbn:ch:bel-1675781 SNL 856 42 http://www.e-helvetica.nb.admin.ch/directAccess?callnumber=bel-1675781 NEBIS 100 1- Chen Long 1988- (DE-588)1166890090 Verfasser aut NEBIS 700 1- Jackson Andrew (DE-588)101733367X Akademischer Betreuer dgs NEBIS 700 1- Noir Jérõme André Roland (DE-588)1089683162 Akademischer Betreuer dgs