Schober Jennifer, Schleicher Dominik, Bovino Stefano, Klessen Ralf S
Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Überle-Strasse 2, D-69120 Heidelberg, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Dec;86(6 Pt 2):066412. doi: 10.1103/PhysRevE.86.066412. Epub 2012 Dec 26.
The present-day Universe is highly magnetized, even though the first magnetic seed fields were most probably extremely weak. To explain the growth of the magnetic field strength over many orders of magnitude, fast amplification processes need to operate. The most efficient mechanism known today is the small-scale dynamo, which converts turbulent kinetic energy into magnetic energy leading to an exponential growth of the magnetic field. The efficiency of the dynamo depends on the type of turbulence indicated by the slope of the turbulence spectrum v(ℓ)∝ℓ^{ϑ}, where v(ℓ) is the eddy velocity at a scale ℓ. We explore turbulent spectra ranging from incompressible Kolmogorov turbulence with ϑ=1/3 to highly compressible Burgers turbulence with ϑ=1/2. In this work, we analyze the properties of the small-scale dynamo for low magnetic Prandtl numbers Pm, which denotes the ratio of the magnetic Reynolds number, Rm, to the hydrodynamical one, Re. We solve the Kazantsev equation, which describes the evolution of the small-scale magnetic field, using the WKB approximation. In the limit of low magnetic Prandtl numbers, the growth rate is proportional to Rm^{(1-ϑ)/(1+ϑ)}. We furthermore discuss the critical magnetic Reynolds number Rm_{crit}, which is required for small-scale dynamo action. The value of Rm_{crit} is roughly 100 for Kolmogorov turbulence and 2700 for Burgers. Furthermore, we discuss that Rm_{crit} provides a stronger constraint in the limit of low Pm than it does for large Pm. We conclude that the small-scale dynamo can operate in the regime of low magnetic Prandtl numbers if the magnetic Reynolds number is large enough. Thus, the magnetic field amplification on small scales can take place in a broad range of physical environments and amplify week magnetic seed fields on short time scales.
如今的宇宙具有很强的磁性，尽管最初的磁种子场很可能极其微弱。为了解释磁场强度在多个数量级上的增长，需要有快速放大过程发挥作用。当今已知最有效的机制是小尺度发电机效应，它将湍动能转化为磁能，导致磁场呈指数增长。发电机效应的效率取决于由湍流谱斜率(v(ℓ)∝ℓ^{ϑ})所表征的湍流类型，其中(v(ℓ))是尺度为(ℓ)时的涡旋速度。我们探索了从不可压缩的科尔莫戈罗夫湍流（(ϑ = 1/3)）到高度可压缩的伯格斯湍流（(ϑ = 1/2)）的湍流谱。在这项工作中，我们分析了低磁普朗特数(Pm)（它表示磁雷诺数(Rm)与流体动力学雷诺数(Re)的比值）情况下小尺度发电机效应的特性。我们使用WKB近似求解描述小尺度磁场演化的卡赞采夫方程。在低磁普朗特数的极限情况下，增长率与(Rm^{(1 - ϑ)/(1 + ϑ)})成正比。我们还讨论了小尺度发电机效应所需的临界磁雷诺数(Rm_{crit})。对于科尔莫戈罗夫湍流，(Rm_{crit})的值约为100，对于伯格斯湍流则为2700。此外，我们还讨论了(Rm_{crit})在低(Pm)极限下比在高(Pm)时提供了更强的约束。我们得出结论，如果磁雷诺数足够大，小尺度发电机效应可以在低磁普朗特数 regime 下发挥作用。因此，小尺度上的磁场放大可以在广泛的物理环境中发生，并在短时间尺度上放大微弱的磁种子场。
