Direct Statistical Simulation (DSS) solves the equations of motion for the statistics of turbulent flows in place of the traditional route of accumulating statistics by Direct Numerical Simulation (DNS). That low-order statistics usually evolve slowly compared with instantaneous dynamics is one important advantage of DSS. Depending on the symmetry of the problem and the choice of averaging operation, however, DSS is usually more expensive computationally than DNS because even low order statistics typically have higher dimension than the underlying fields. Here we show that it is possible to go much further by using Proper Orthogonal Decomposition (POD) to address the "curse of dimensionality." We apply POD directly to DSS in the form of expansions in the equal-time cumulants to second order (CE2). We explore two averaging operations (zonal and ensemble) and test the approach on two idealized barotropic models on a rotating sphere (a jet that relaxes deterministically towards an unstable profile, and a stochastically-driven flow that spontaneously organizes into jets). Order-of-magnitude savings in computational cost are obtained in the reduced basis, potentially enabling access to parameter regimes beyond the reach of DNS.

中文翻译：

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直接统计模拟的降维

直接统计模拟 (DSS) 解决了湍流统计的运动方程，代替了通过直接数值模拟 (DNS) 累积统计的传统途径。与瞬时动态相比，低阶统计通常发展缓慢是 DSS 的一项重要优势。然而，根据问题的对称性和平均操作的选择，DSS 在计算上通常比 DNS 更昂贵，因为即使是低阶统计数据通常也比基础字段具有更高的维度。在这里，我们表明可以通过使用适当的正交分解 (POD) 来解决“维数灾难”，从而走得更远。我们将 POD 以等时累积量扩展到二阶 (CE2) 的形式直接应用于 DSS。我们探索了两个平均操作（区域和集合），并在旋转球体上的两个理想化正压模型上测试该方法（一个射流确定性地向不稳定剖面松弛，以及一个随机驱动的流动，该流自发地组织成射流）。计算成本的数量级节省是在减少的基础上获得的，可能允许访问超出 DNS 范围的参数机制。