While the saturated magnetic energy is independent of viscosity in dynamo experiments, it remains viscosity dependent in state-of-the-art 3D direct numerical simulations (DNS). Extrapolating such viscous scaling laws to realistic parameter values leads to an underestimation of the magnetic energy by several orders of magnitude. The origin of this discrepancy is that fully 3D DNS cannot reach low enough values of the magnetic Prandtl number Pm. To bypass this limitation and investigate dynamo saturation at very low Pm, we focus on the vicinity of the dynamo threshold in a rapidly rotating flow: the velocity field then depends on two spatial coordinates only, while the magnetic field consists of a single Fourier mode in the third direction. We perform numerical simulations of the resulting set of reduced equations for Pm down to $2\times {10}^{-5}$. This parameter regime is currently out of reach to fully 3D DNS. We show that the magnetic energy transitions from a high-Pm viscous scaling regime to a low-Pm turbulent scaling regime, the latter being independent of viscosity. The transition to the turbulent saturation regime occurs at a low value of the magnetic Prandtl number, $\mathrm{Pm}\simeq {10}^{-3}$, which explains why it has been overlooked by numerical studies so far.

中文翻译：

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过渡到湍流发电机饱和

尽管在发电机实验中饱和磁能与粘度无关，但在最新的3D直接数值模拟（DNS）中，饱和磁能仍然取决于粘度。将这种粘性缩放定律外推到实际参数值会导致磁能低估几个数量级。这种差异的根源在于，完整的3D DNS无法达到足够低的磁Prandtl数Pm值。为了绕过此限制并研究在非常低的Pm下的发电机饱和度，我们将重点放在快速旋转的流中发电机阈值附近：速度场仅取决于两个空间坐标，而磁场仅由一个傅里叶模式构成。第三方向。我们对最终的Pm简化方程组进行数值模拟$\mathrm{2个}\times {10}^{-5}$。目前，此参数范围对于完全3D DNS而言是遥不可及的。我们表明，磁能从高Pm粘性结垢状态过渡到低Pm湍流结垢状态，后者与粘度无关。向湍流饱和状态的过渡发生在磁Prandtl值较低的情况下，$\mathrm{下午}\simeq {10}^{-3}$，这解释了为什么到目前为止，数值研究都忽略了它。