Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos.
Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos.
复制标题
DOI：
10.1103/physreve.79.016205
复制
发表时间：
2008-06
期刊：
影响因子：
--
通讯作者：
Oleg N. Kirillov;U. Günther;F. Stefani
中科院分区：
文献类型：
--
作者：
Oleg N. Kirillov;U. Günther;F. Stefani
利用同伦族边界特征值问题求解均值场{2}发电机具有螺旋湍流参数α (r)= α _{0}+gammaDeltaalpha(r)和同伦参数β[0,1]，我们证明了Dirichlet（理想，β =0）边界条件下的潜在的漩涡点网络实质上决定了特征值的排列，从而决定了Robin（物理现实，β =1）边界条件下发电机不稳定性的特征。在（α _{0}, β, γ）空间中，β =1时振荡解的阿诺德舌结束于β =0时的恶魔点。在尖锥点附近，一阶近似锥体的三维舌形的空间取向由锥体顶点处的尖锥交叉所涉及的模态的Krein特征决定。Krein空间诱导的共振区几何解释了寻找导致谱异常点的α剖面的微妙之处，这是最近地磁场极性反转理论的重要组成部分。
Using a homotopic family of boundary eigenvalue problems for the mean-field alpha;{2} dynamo with helical turbulence parameter alpha(r)=alpha_{0}+gammaDeltaalpha(r) and homotopy parameter beta[0,1] , we show that the underlying network of diabolical points for Dirichlet (idealized, beta=0 ) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for Robin (physically realistic, beta=1 ) boundary conditions. In the (alpha_{0},beta,gamma) space the Arnold tongues of oscillatory solutions at beta=1 end up at the diabolical points for beta=0 . In the vicinity of the diabolical points the space orientation of the three-dimensional tongues, which are cones in first-order approximation, is determined by the Krein signature of the modes involved in the diabolical crossings at the apexes of the cones. The Krein space-induced geometry of the resonance zones explains the subtleties in finding alpha profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.