Title: Numerical simulations of a stochastic dynamics leading to cascades and loss of regularity: applications to fluid turbulence and generation of fractional Gaussian fields Show Chinese title

Title: 随机动力学的数值模拟导致级联和正则性的丧失：流体湍流和分数高斯场生成的应用

Abstract: Motivated by the modeling of the spatial structure of the velocity field of three-dimensional turbulent flows, and the phenomenology of cascade phenomena, a linear dynamics has been recently proposed able to generate high velocity gradients from a smooth-in-space forcing term. It is based on a linear Partial Differential Equation (PDE) stirred by an additive random forcing term which is delta-correlated in time. The underlying proposed deterministic mechanism corresponds to a transport in Fourier space which aims at transferring energy injected at large scales towards small scales. The key role of the random forcing is to realize these transfers in a statistically homogeneous way. Whereas at finite times and positive viscosity the solutions are smooth, a loss of regularity is observed for the statistically stationary state in the inviscid limit. We here present novel simulations, based on finite volume methods in the Fourier domain and a splitting method in time, which are more accurate than the pseudo-spectral simulations. We show that the novel algorithm is able to reproduce accurately the expected local and statistical structure of the predicted solutions. We conduct numerical simulations in one, two and three spatial dimensions, and we display the solutions both in physical and Fourier spaces. We additionally display key statistical quantities such as second-order structure functions and power spectral densities at various viscosities. Show Chinese abstract

Abstract: 受三维湍流速度场的空间结构建模以及级联现象的物理现象的启发，最近提出了一种线性动力学，能够从空间平滑的强迫项中产生高速度梯度。 它基于一个由时间delta相关随机强迫项驱动的线性偏微分方程（PDE）。所提出的确定性机制对应于傅里叶空间中的传输，旨在将大尺度注入的能量转移到小尺度。 随机强迫的关键作用是统计上均匀地实现这些转移。 尽管在有限时间和正粘性下解是光滑的，但在无粘性极限下的统计稳态中观察到了正则性的损失。 我们这里展示了基于傅里叶域中的有限体积方法和时间分裂方法的新模拟，比伪谱模拟更准确。 我们表明，新算法能够准确再现预测解的预期局部和统计结构。 我们在一维、二维和三维空间中进行了数值模拟，并在物理空间和傅里叶空间中展示了解。 我们还展示了不同粘度下的关键统计量，如二阶结构函数和功率谱密度。