统计发电机理论：模式激励
Statistical dynamo theory: Mode excitation
摘要：
本文计算了任意形状发电机产生的磁场的最低阶多极系数的统计特性。为此，我们将该域展开为一个完备的双正交基函数集，即B = sum_k a^k（t）b^k（r）。我们考虑一个线性问题，假设流体流动的统计性质是已知的。湍流对流可以具有任意的空间尺度分布。展开系数a^k（t）的时间演化由随机微分方程控制，我们从该方程推断它们的平均值<a^k>、自相关函数<a^k（t）a^{k*}（t+tau）>和互相关方程<a^k a^l*>。发电机方程的本征函数（具有本征值λ_k）是一个优选的集合，根据这个集合，我们的结果呈现出最简单的形式。发电机的磁场由瞬态激发的本征模组成，其频率和相干时间分别由Im（λ k）和-1/（Re λ k）给出.相对均方根本征模的激发水平以及因此磁能在空间尺度上的分布由线性理论确定。一个表达式是为<|a^k| ^2> / <|a^0| ^2>在基模b^0具有主振幅的情况下，我们概述了如何计算该表达式。据估计，|a^k| ^2>/<|a^0| ^2> ~ 1/N其中N是发电机中对流单体的数量。我们发现，一个短的相关时间（或FOSA）的老问题已部分消除。最后，我们证明了对于一个简单的统计稳定的有限电阻率发电机，其所有特征值都服从Re（λ_k）< 0。
英文摘要：
We compute statistical properties of the lowest-order multipole coefficients of the magnetic field generated by a dynamo of arbitrary shape. To this end we expand the field in a complete biorthogonal set of base functions, viz. B = sum_k a^k(t) b^k(r). We consider a linear problem and the statistical properties of the fluid flow are supposed to be given. The turbulent convection may have an arbitrary distribution of spatial scales. The time evolution of the expansion coefficients a^k(t) is governed by a stochastic differential equation from which we infer their averages <a^k>, autocorrelation functions <a^k(t) a^{k*}(t+tau)>, and an equation for the cross correlations <a^k a^l*>. The eigenfunctions of the dynamo equation (with eigenvalues lambda_k) turn out to be a preferred set in terms of which our results assume their simplest form. The magnetic field of the dynamo is shown to consist of transiently excited eigenmodes whose frequency and coherence time is given by Im(lambda_k) and -1/(Re lambda_k), respectively. The relative r.m.s. excitation level of the eigenmodes, and hence the distribution of magnetic energy over spatial scales, is determined by linear theory. An expression is derived for <|a^k|^2> / <|a^0|^2> in case the fundamental mode b^0 has a dominant amplitude, and we outline how this expression may be evaluated. It is estimated that <|a^k|^2>/<|a^0|^2> ~ 1/N where N is the number of convective cells in the dynamo. We show that the old problem of a short correlation time (or FOSA) has been partially eliminated. Finally we prove that for a simple statistically steady dynamo with finite resistivity all eigenvalues obey Re(lambda_k) < 0.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
