Feudel F, Gellert M, Rüdiger S, Witt A, Seehafer N
Institut für Physik, Universität Potsdam, PF 601553, D-14415 Potsdam, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Oct;68(4 Pt 2):046302. doi: 10.1103/PhysRevE.68.046302. Epub 2003 Oct 8.
The Roberts flow, a helical flow in the form of convectionlike rolls, is known to be capable of both kinematic and nonlinear dynamo action. We study the Roberts dynamo with particular attention being paid to the spatial structure of the generated magnetic field and its back-reaction on the flow. The dynamo bifurcation is decisively determined by the symmetry group of the problem, which is given by a subgroup of discrete transformations and a continuous translational invariance of the flow. In the bifurcation the continuous symmetry is broken while the discrete subgroup symmetry completely survives. Its actions help in understanding the spatial structures of the magnetic field and of the modified flow. In accordance with experimental observations, the magnetic field component perpendicular to the originally invariant direction is much stronger than the component in this direction. Furthermore, the magnetic field is largely concentrated in layers separating the convectionlike rolls of the flow and containing, in particular, its stagnation points, which are isolated for the modified flow while they are line filling for the original Roberts flow. The magnetic field is strongest near beta-type stagnation points, with a two-dimensional unstable and a one-dimensional stable manifold, and is weak near alpha-type stagnation points, with a two-dimensional stable and a one-dimensional unstable manifold. This contrasts with the usual picture that dynamo action is promoted at the alpha points and impeded at the beta points. Both the creation of isolated stagnation points and the concentration of strong fields at the beta points may be understood as a result of the way in which the Roberts dynamo saturates. It is also found that, while the original Roberts flow is regular, the modified flow is chaotic in the layers between the convectionlike rolls where the magnetic field is concentrated. This chaoticity, which results from the back-reaction of the magnetic field on the flow, appears to merely enhance magnetic diffusion rather than to strengthen the dynamo effect.
罗伯茨流是一种呈对流状涡旋形式的螺旋流，已知它既能产生运动学发电机作用，也能产生非线性发电机作用。我们研究罗伯茨发电机，特别关注所产生磁场的空间结构及其对流动的反作用。发电机分岔由问题的对称群决定性地决定，该对称群由离散变换的一个子群和流动的连续平移不变性给出。在分岔过程中，连续对称性被打破，而离散子群对称性完全保留。其作用有助于理解磁场和修正后流动的空间结构。根据实验观测，垂直于原始不变方向的磁场分量比该方向上的分量强得多。此外，磁场主要集中在分隔流动的对流状涡旋的层中，特别是包含其驻点，这些驻点对于修正后的流动是孤立的，而对于原始的罗伯茨流是线填充的。磁场在具有二维不稳定和一维稳定流形的β型驻点附近最强，在具有二维稳定和一维不稳定流形的α型驻点附近较弱。这与通常认为发电机作用在α点促进而在β点受阻的情况形成对比。孤立驻点的产生以及强场在β点的集中都可以理解为罗伯茨发电机饱和方式的结果。还发现，虽然原始的罗伯茨流是规则的，但在磁场集中的对流状涡旋之间的层中，修正后的流动是混沌的。这种由磁场对流动的反作用导致的混沌性似乎只是增强了磁扩散，而不是增强了发电机效应。
