Analytic solution of an oscillatory migratory alpha^2 stellar dynamo
Analytic solution of an oscillatory migratory alpha^2 stellar dynamo
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振荡迁移 α^2 恒星发电机的解析解
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发表时间：
2016
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通讯作者：
A. Brandenburg
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作者：
A. Brandenburg
平均场感应方程的解析解可预测无界或周期域中均匀螺旋湍流或恒定 α 效应的非振荡发电机。通常认为振荡发电机对于恒定的阿尔法是不可能的。我们提出了一个一维有界域的解析解，产生常数 α 的振荡解，但两端的边界条件不同（狄利克雷和冯·诺依曼或完美导体和真空）。我们求解二阶复方程并将两个独立的解叠加以遵守两个边界条件。该解决方案具有与时间无关的能量密度。在函数值消失的一端，二阶导数是有限的，在节点与拐点重合的情况下，不能用类正弦展开函数正确地再现二阶导数。所获得的解决方案可以作为数值发电机实验的基准，并作为教学说明，说明对于具有恒定 α 的发电机，振荡发电机是可能的。
Analytic solutions of the mean-field induction equation predict a nonoscillatory dynamo for uniform helical turbulence or constant alpha effect in unbounded or periodic domains. Oscillatory dynamos are generally thought impossible for constant alpha. We present an analytic solution for a one-dimensional bounded domain resulting in oscillatory solutions for constant alpha, but different (Dirichlet and von Neumann or perfect conductor and vacuum) boundary conditions on the two ends. We solve a second order complex equation and superimpose two independent solutions to obey both boundary conditions. The solution has time-independent energy density. On one end where the function value vanishes, the second derivative is finite, which would not be correctly reproduced with sine-like expansion functions where a node coincides with an inflection point. The obtained solution may serve as a benchmark for numerical dynamo experiments and as a pedagogical illustration that oscillatory dynamos are possible for dynamos with constant alpha.