Kinematic dynamo onset and magnetic field saturation in rotating spherical Couette and periodic box simulations
Kinematic dynamo onset and magnetic field saturation in rotating spherical Couette and periodic box simulations
by Konstantin Finke
Date of Examination:2013-06-19
Date of issue:2013-07-26
Advisor:Prof. Dr. Andreas Tilgner
Referee:Prof. Dr. Andreas Tilgner
Referee:Prof. Dr. Christensen Ulrich
Persistent Address:
http://dx.doi.org/10.53846/goediss-3963
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Abstract
English
Magnetic fields are ubiquitous in the universe and can be found in celestial bodies, galaxies, stars including our Sun and planets like the Earth or Jupiter. Due to the fact that at least in the Earth's interior, temperatures are well above the Curie temperature, its magnetic field cannot result from permanent magnetisation. Moreover, its time dependence gives rise to the assumption, that the generation of the magnetic field must be the result of a very complex dynamical process. The idea is that the magnetic fields can be sustained by self-inductive processes of a moving electrically conducting fluid or plasma. Today, it is generally accepted that this magnetohydrodynamic (MHD) dynamo effect is responsible for the magnetic field generation in most stellar objects. This PhD thesis consists of two parts which treat two fundamental aspects in dynamo theory. The first part focuses on the kinematic dynamo threshold of the spherical Couette flow and is related with the liquid sodium experiment in Maryland. Since previous work on this system failed to come up with promising results for spherical Couette experiments to succeed in creating a dynamo, the simulations are repeated in order to compare the results with a different driving mechanism. The spherical Couette system is driven by the moving boundaries (in this case, only the inner sphere is rotating), which are coupled to the fluid by viscous drag. Compared to the Maryland experiment this is represented by smooth boundaries. Since the boundary layer is dependent on the rotation of the inner sphere, the efficiency of the flow could be increased by a driving force that drives the flow in a constant distance from the inner boundary so that it becomes independent from the rotation rate and increases the momentum transfer. Accordingly, the Maryland experiment can be modified by rough inner boundaries or blades attached to the inner sphere. It is shown that by implementing rough boundaries the flow becomes more efficient to dynamo action and the conditions to sustain a dynamo are improved significantly. The second part deals with the saturation mechanism of a magnetic field in a rotating system. Therefore, the saturation of the magnetic field within a G. O. Roberts like driven flow in a rotating frame of reference is investigated. The flow structure in such celestial bodies is believed to have a similar shape. Once a small magnetic field rises within a flow, at a certain point its Lorentz force reaches a strength comparable to the driving force and reorganises the flow, so that the magnetic field saturates. This part focuses on the effect of the rotation rate on the saturation mechanism of the dynamo and the reorganisation of the velocity field by the Lorentz force. As long as the parameters are close to the kinematic dynamo onset, the Lorentz force is small and can be treated as a weakly non-linear perturbation in the mean field picture. The approximated MHD equations are solved analytically in a rotating periodic box, where the flow is driven by a force field corresponding to the G.O. Roberts flow. In order to test whether these analytical assumptions are reliable, numerical simulations of the full MHD equation are performed. Basic results of the numerical simulations could be recovered by the analytical approach. This includes the dependency of the magnetic energy on the control parameters Rm and Ek. Additionally the mechanism and the interactions of the flow and the magnetic field could be claryfied by this model. The only aspect which could not be explained sufficiently is the perturbation order, in which the rotational dependent term appears.
Keywords: Spherical Couette; G. O. Roberts flow; Dynamo; Turbulence; Weakly non-linear theory