Luo Jiawen, Chen Long, Li Kuan, Jackson Andrew
Institut für Geophysik, ETH Zürich, Sonneggstrasse 5, 8092 Zürich, Switzerland.
Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Road, Durham DH1 3LE, UK.
Proc Math Phys Eng Sci. 2020 Jan;476(2233):20190675. doi: 10.1098/rspa.2019.0675. Epub 2020 Jan 8.
A variational optimization approach is used to optimize kinematic dynamos in a unit sphere and locate the enstrophy-based critical magnetic Reynolds number for dynamo action. The magnetic boundary condition is chosen to be either pseudo-vacuum or perfectly conducting. Spectra of the optimal flows corresponding to these two magnetic boundary conditions are identical since theory shows that they are relatable by reversing the flow field (Favier & Proctor 2013 , 031001 (doi:10.1103/physreve.88.031001)). A no-slip boundary for the flow field gives a critical magnetic Reynolds number of 62.06, while a free-slip boundary reduces this number to 57.07. Optimal solutions are found to possess certain rotation symmetries (or anti-symmetries) and optimal flows share certain common features. The flows localize in a small region near the sphere's centre and spiral upwards with very large velocity and vorticity, so that they are locally nearly Beltrami. We also derive a new lower bound on the magnetic Reynolds number for dynamo action, which, for the case of enstrophy normalization, is five times larger than the previous best bound.
采用变分优化方法对单位球体内的运动发电机进行优化，并确定基于涡旋度的发电机作用临界磁雷诺数。磁边界条件选择为伪真空或完全导电。对应于这两种磁边界条件的最优流的谱是相同的，因为理论表明通过反转流场它们是相关的（法维耶与普罗克特，2013 年，031001（doi:10.1103/physreve.88.031001））。流场的无滑移边界给出的临界磁雷诺数为62.06，而自由滑移边界将此数减小到57.07。发现最优解具有某些旋转对称性（或反对称性），并且最优流具有某些共同特征。这些流集中在球体中心附近的一个小区域内，并以非常大的速度和涡度向上螺旋，使得它们在局部上近似为贝尔特拉米流。我们还推导了发电机作用磁雷诺数的一个新的下限，对于涡旋度归一化的情况，该下限比之前的最佳下限大五倍。
