It is a challenge, to efficiently and accurately numerical model atmospheric and oceanic flows with strong interactions among eddies of different scales. Traditionally the most efficient methods involve finite differences, but the most accurate are the spectral methods. Small eddies are inefficiently represented by spectral methods, while derivatives are relatively inaccurate using finite differences. Nonlinearities present further difficulties. This presentation will review the Spectral Element Method (SEM), including past applications to atmospheric and oceanic simulation and a very recent generalization to Multiresolution Adaptive Spectral Elements (MASE). SEM decomposes the computational domain into finite elements, thus achieving the efficiency of finite differences. However within each element, Gaussian quadrature is used in integral formulations of the dynamics, thus achieving the accuracy of spectral methods. MASE adds to these features, the ability to adaptively refine the resolution to eddies or other dynamically emerging structures of arbitrary size, while maintaining high accuracy and efficiency throughout the domain.

Recent SEM results include modeling the breakdown of the polar stratospheric vortex, at T363. The MASE method will be demonstrated on 2D fluid dynamics test cases that generate a wide range of strongly interacting scales, such as the 2D Burgers Equation and the Shallow Water Equations.