Turbulent fluid flows in atmospheric and oceanic sciences are characterized by strongly transient features with spatial inhomogeneity, spanning a wide range of spatial and temporal scales. While large-scale dynamics are often well approximated by closure schemes there is still a need to efficiently represent the corresponding small-scale features, when it comes to the risk analysis for extreme events. We introduce a data-driven framework for the stochastic parametrization of the small spatial scales in terms of the large ones. The framework employs a spherical wavelet decomposition to partition field quantities, obtained from reanalysis data into non-overlapping spectral components. Using these time-series we formulate, for each spatial location, a machine-learning scheme that naturally ‘splits’ the small-scales into a predictable part, which can be effectively parametrized in terms of the large-scales time-series, and a stochastic residual, which cannot be uniquely determined using the large-scale information. The later is represented using a conditionally Gaussian process, a choice that allows us to overcome the need for a vast amount of training data, which for climate applications, is naturally limited to a single realization for each spatial location. Using a second round of machine-learning we parametrize, for each location, the covariance of the stochastic component in terms of the large scales. We employ the machine-learned statistics to parsimoniously reconstruct random realizations of the small scales. We demonstrate the approach on reanalysis data involving vorticity over Western Europe and we show that the reconstructed random samples for the small scales result in excellent agreement to the spatial spectrum, single-point probability density functions, and temporal spectral content.

大气和海洋科学中的湍流流体具有强烈的瞬态特征和空间不均匀性，跨越广泛的空间和时间尺度。虽然大规模动态通常可以通过闭包方案很好地近似，但在极端事件的风险分析中，仍然需要有效地表示相应的小规模特征。我们引入了一个数据驱动的框架，用于根据大空间尺度对小空间尺度进行随机参数化。该框架采用球面小波分解将从再分析数据中获得的场量划分为不重叠的光谱分量。使用这些时间序列，我们为每个空间位置制定了一个机器学习方案，可以自然地将小尺度“拆分”为可预测的部分，可以在大规模时间序列和随机残差方面有效地参数化，而随机残差不能使用大规模信息唯一确定。后者使用条件高斯过程表示，这种选择使我们能够克服对大量训练数据的需求，而对于气候应用，这些数据自然仅限于每个空间位置的单一实现。使用第二轮机器学习，我们对每个位置的随机分量在大尺度方面的协方差进行参数化。我们采用机器学习的统计数据来简约地重建小尺度的随机实现。