振荡和非振荡α^2-发电机的谱性质
Spectral properties of oscillatory and non-oscillatory α^2-dynamos
摘要：
本文计算了各向同性α效应在不同空间分布下线性α^{2}发电机的本征值和本征函数。当α在径向上有符号变化时，得到了振动解.时间相关的解决方案出现在所谓的例外点，两个固定模式合并，并继续作为一个振荡的本征函数共轭复特征值。振荡和非振荡的解决方案的接近可以作为反转模型，描述突然的极性开关的偶极子引起的噪声的基本成分。而具有不同的磁扩散率的内核的存在下，占主导地位的发电机本征模式的字符有显着的影响不大，赤道对称性破缺的引入显着改变的几何特征的解决方案。在发电机阈值附近，即使对称破缺很小，主导模式也对应于半球发电机。这种行为可以解释为近似的偶极-四极退化的未扰动问题。在α和β效应的更现实的各向异性或通过由磁场的反向反应（磁淬灭）引起的非线性的情况下，可能发生更复杂的情况。
英文摘要：
The eigenvalues and eigenfunctions of a linear α^{2}-dynamo have been computed for different spatial distributions of an isotropic α-effect. Oscillatory solutions are obtained when αexhibits a sign change in the radial direction. The time-dependent solutions arise at so called exceptional points where two stationary modes merge and continue as an oscillatory eigenfunction with conjugate complex eigenvalues. The close proximity of oscillatory and non-oscillatory solutions may serve as the basic ingredient for reversal models that describe abrupt polarity switches of a dipole induced by noise. Whereas the presence of an inner core with different magnetic diffusivity has remarkable little impact on the character of the dominating dynamo eigenmodes, the introduction of equatorial symmetry breaking considerably changes the geometric character of the solutions. Around the dynamo threshold the leading modes correspond to hemispherical dynamos even when the symmetry breaking is small. This behavior can be explained by the approximate dipole-quadrupole degeneration for the unperturbed problem. More complicated scenarios may occur in case of more realistic anisotropies of α- and β-effect or through non-linearities caused by the back-reaction of the magnetic field (magnetic quenching).
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