We study the Lagrangian trajectories of statistically isotropic, homogeneous, and stationary divergence-free spatiotemporal random vector fields. We design this advecting Eulerian velocity field such that it gets asymptotically rough and multifractal, both in space and time, as it is demanded by the phenomenology of turbulence at infinite Reynolds numbers. We then solve numerically the flow equations for a differentiable version of this field. We observe that trajectories get also rough, characterized by nearly the same Hurst exponent as the one of our prescribed advecting field. Moreover, even when considering the simplest situation of the advection by a fractional Gaussian field, we evidence in the Lagrangian framework additional intermittent corrections. The present approach involves properly defined random fields, and asks for a rigorous treatment that would explain our numerical findings and deepen our understanding of this long lasting problem.

中文翻译：

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时空湍流状随机场的流动。

我们研究了统计各向同性，齐次和平稳无散度的时空随机矢量场的拉格朗日轨迹。我们设计这种平缓的欧拉速度场，使其在空间和时间上渐近粗糙且具有多重分形，这是在无限雷诺数下的湍流现象学所要求的。然后，我们以数字方式求解该字段可微分形式的流动方程。我们观察到，轨迹也变得粗糙，其特征是与我们规定的平流场之一几乎相同的赫斯特指数。此外，即使考虑分数分数高斯场对流的最简单情况，我们也在拉格朗日框架中证明了其他间歇性校正。本方法涉及正确定义的随机字段，