Giesecke André, Stefani Frank, Burguete Javier
Institute of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf, P. O. Box 510119, D-01314 Dresden, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Dec;86(6 Pt 2):066303. doi: 10.1103/PhysRevE.86.066303. Epub 2012 Dec 7.
We present numerical simulations of the kinematic induction equation in order to examine the dynamo efficiency of an axisymmetric von Kármán-like flow subject to time-dependent nonaxisymmetric velocity perturbations. The numerical model is based on the setup of the French von Kármán-sodium dynamo (VKS) and on the flow measurements from a water experiment conducted at the University of Navarra in Pamplona, Spain. The principal experimental observations that are modeled in our simulations are nonaxisymmetric vortexlike structures which perform an azimuthal drift motion in the equatorial plane. Our simulations show that the interactions of these periodic flow perturbations with the fundamental drift of the magnetic eigenmode (including the special case of nondrifting fields) essentially determine the temporal behavior of the dynamo state. We find two distinct regimes of dynamo action that depend on the (prescribed) drift frequency of an (m=2) vortexlike flow perturbation. For comparatively slowly drifting vortices we observe a narrow window with enhanced growth rates and a drift of the magnetic eigenmode that is synchronized with the perturbation drift. The resonance-like enhancement of the growth rates takes place when the vortex drift frequency roughly equals the drift frequency of the magnetic eigenmode in the unperturbed system. Outside of this small window, the field generation is hampered compared to the unperturbed case, and the field amplitude of the magnetic eigenmode is modulated with approximately twice the vortex drift frequency. The abrupt transition between the resonant regime and the modulated regime is identified as a spectral exceptional point where eigenvalues (growth rates and frequencies) and eigenfunctions of two previously independent modes collapse. In the actual configuration the drift frequencies of the velocity perturbations that are observed in the water experiment are much larger than the fundamental drift frequency of the magnetic eigenmode that is obtained from our numerical simulations. Hence, we conclude that the fulfillment of the resonance condition might be unlikely in present day dynamo experiments. However, a possibility to increase the dynamo efficiency in the VKS experiment might be realized by an application of holes or fingers on the outer boundary in the equatorial plane. These mechanical distortions provoke an anchorage of the vortices at fixed positions thus allowing an adjustment of the temporal behavior of the nonaxisymmetric flow perturbations.
我们给出了运动感应方程的数值模拟，以研究轴对称类冯·卡门流在随时间变化的非轴对称速度扰动作用下的发电机效率。数值模型基于法国冯·卡门 - 钠发电机（VKS）的设置以及西班牙潘普洛纳纳瓦拉大学进行的水实验中的流量测量。我们模拟中建模的主要实验观测结果是非轴对称涡旋状结构，它们在赤道平面内进行方位漂移运动。我们的模拟表明，这些周期性流动扰动与磁本征模的基本漂移（包括非漂移场的特殊情况）之间的相互作用基本上决定了发电机状态的时间行为。我们发现了两种不同的发电机作用模式，这取决于（规定的）（m = 2）涡旋状流动扰动的漂移频率。对于相对缓慢漂移的涡旋，我们观察到一个具有增强增长率的窄窗口以及与扰动漂移同步的磁本征模漂移。当涡旋漂移频率大致等于未受扰动系统中磁本征模的漂移频率时，增长率会出现类似共振的增强。在这个小窗口之外，与未受扰动的情况相比，场的产生受到阻碍，并且磁本征模的场振幅以大约两倍的涡旋漂移频率进行调制。共振模式和调制模式之间的突然转变被确定为一个光谱例外点，在此处两个先前独立模式的特征值（增长率和频率）以及特征函数会合并。在实际配置中，水实验中观测到的速度扰动的漂移频率远大于我们数值模拟得到的磁本征模的基本漂移频率。因此，我们得出结论，在当今的发电机实验中不太可能满足共振条件。然而，通过在赤道平面的外边界应用孔洞或指状物，有可能在VKS实验中提高发电机效率。这些机械变形会使涡旋固定在特定位置，从而允许调整非轴对称流动扰动的时间行为。
