We present a semi-Lagrangian characteristic mapping method for the incompressible Euler equations on a rotating sphere. The numerical method uses a spatio-temporal discretization of the inverse flow map generated by the Eulerian velocity as a composition of sub-interval flows formed by $C^1$ spherical spline interpolants. This approximation technique has the capacity of resolving sub-grid scales generated over time without increasing the spatial resolution of the computational grid. The numerical method is analyzed and validated using standard test cases yielding third-order accuracy in the supremum norm. Numerical experiments illustrating the unique resolution properties of the method are performed and demonstrate the ability to reproduce the forward energy cascade at sub-grid scales by upsampling the numerical solution.

翻译：我们提出了一种适用于旋转球体上不可压缩欧拉方程的半拉格朗日特性映射方法。该数值方法利用欧拉速度生成的反向流映射的时空离散化，通过由$C^1$球面样条插值函数构成的子区间流复合实现。这种近似技术能够在无需增加计算网格空间分辨率的情况下，解析随时间演化的亚网格尺度。通过标准测试案例对数值方法进行了分析与验证，结果表明其在最大模下具有三阶精度。数值实验展示了该方法独特的解析特性，并证明了通过上采样数值解能够重现亚网格尺度的正向能量级联过程。