Fast and Accurate Computation of Vertical Modes

摘要

The vertical modes of linearized equations of motion are widely used by the oceanographic community in numerous theoretical and observational contexts. However, the standard approach for solving the generalized eigenvalue problem using second‐order finite difference matrices produces errors for all but the few lowest modes, and increasing resolution quickly becomes too slow as the computational complexity of eigenvalue algorithms increases as . Existing methods are therefore inadequate for computing a full spectrum of internal waves, such as needed for initializing a numerical model with a full internal wave spectrum. Here we show that rewriting the eigenvalue problem in stretched coordinates and projecting onto Chebyshev polynomials results in substantially more accurate modes than finite differencing at a fraction of the computational cost. We also compute the surface quasigeostrophic modes using the same methods. All spectral and finite difference algorithms are made available in a suite of Matlab classes that have been validated against known analytical solutions in constant and exponential stratification. Plain Language Summary The ocean is filled with internal waves that primarily move fluid back and forth horizontally and, to a lesser degree, vertically. The amount of vertical motion for each wave depends on depth because the fluid density in the ocean also changes with depth. The density profile of the ocean varies both seasonally and geographically, which means that vertical structure of each wave is also a function of season and geographic location. In order to diagnose the sizes of the internal waves, scientists need to accurately compute the motion predicted for each wave from a given density profile. The standard methodology for computing the vertical structure of the fluid motion only works well for the biggest and slowest moving waves. This paper introduces a different methodology which is both computationally faster and more accurate than the traditional methods. All of the code for computing the vertical structure of waves is implemented in Matlab and is freely available. The paper also explores some of the most common errors that will be encountered when using ocean data to diagnose the vertical structure.