Order and Resolution for Computational Ocean Dynamics

Abstract

An ocean flow that has all its scales resolved on a model grid can be more efficiently calculated to within a required accuracy by using high-order numerics than by grid refinement with low-order numerics. The differencing order must be at least as great as the space?time dimensionality D of the model to ensure that grid refinement reduces truncation error at least as quickly as computational cost increases. Ocean flows often have variability on a wide range of scales that cannot all be resolved on any practical grid. In such circumstances the distribution of variability among the scales determines whether grid refinement or increased order results in the greatest accuracy per unit computational cost. A model that simulates the ?5/3 power law of the inertial subrange of three-dimensional turbulence would most efficiently exploit low-order numerics for all terms. The spectra of different terms in the equations of motion can be different and can therefore require different orders of accuracy for efficient computation. Modeling geophysical turbulence with a power law of ?3 would require high-order numerics for the advective terms but low-order numerics would be sufficient for other terms. Output from several ocean models are observed to have spectra that are sufficiently red to justify using high-order numerics for all terms. In the case of one relatively simple ocean modeling problem the author demonstrates that leading-order terms dominate the truncation error.