Inertial Stability and Phase Error of Time Integration Schemes in Ocean General Circulation Models

Abstract

Numerical finite-difference schemes of time integration in widely used ocean general circulation models are systematically examined to ensure the correct and accurate discretization of the Coriolis terms. Two groups of numerical schemes are categorized. One group is suitable for simulating an inertial wave system and geostrophic adjustment processes in the ocean with the necessary condition for stability being |F| = |f| ?t < 1 (where f is the Coriolis parameter and ?t is the integration time step in the model), such as the predictor?corrector scheme (as shown in this study), the most commonly used leapfrog scheme (as used in MICOM, POM, SPEM, and many others), Euler-centered scheme (as used in SOMS), and leapfrog scheme plus Euler-centered Coriolis terms [as used in the Geophysical Fluid Dynamics Laboratory (GFDL) model]. The other group is able to serve as a long-term climate study using a large integration time step that may violate |F| = |f| ?t < 1 by damping out inertial waves, such as the GFDL scheme plus Euler-backward Coriolis terms and the Euler predictor?corrector scheme plus an implicit treatment of the Coriolis terms used in OPYC model. Caution is made regarding the use of the Euler-forward and other schemes that produce unstable inertial waves; this problem could be serious for a calculation longer than one week. The predictor?corrector scheme is recommended as a replacement for the simple Euler-forward scheme. The explicit leapfrog and predictor?corrector schemes tend to overestimate the phase frequency, whereas the Euler schemes and implicit schemes underestimate it. To better simulate the correct phase frequency, F < 0.1 is recommended. Furthermore, an alternate use of an explicit scheme (e.g., leapfrog) and an implicit scheme (e.g., Euler backward or Masuno scheme, etc.) is strongly recommended to preserve the correct phase frequency.