ALBERT

Notes: Abstract: In this paper we study the sheaf of differential operators on a flag manifold X in characteristic p〉0. We generalize the non-vanishing theorem of Haastert on the associated filtration of . We make use of the lifting of X to characteristic zero and the complex geometry of X.

Notes: Abstract Meridional, linear, and free modes of global, primitive-equation, ocean-atmosphere models were analyzed to see if they contain multi-year, especially decadal (∼ 10–30 years), oscillation time scale modes. A two-layer model of the global ocean and a two-level model of the global atmosphere were formulated. Both models were linearized around axially-symmetric basic states containing mean meridional circulations. The linearized perturbation system was solved as an eigenvalue problem. The operator matrix was discretized in the north-south direction with centered finite differences. Uncoupled, meridional modes of oscillation of the ocean and the atmosphere models were calculated. Calculations were performed at three grid spacings (5°, 2.5° and 1.25°) and for two types of basic states (symmetric and asymmetric). Uncoupled, free oceanic modes in the presence of mean meridional circulations have oscillation time scales ranging from two years to several centuries. Such low frequency meridional modes do not exist in the ocean model if there are no mean meridional circulations. A large number of oceanic modes are grouped around decadal oscillation time scales. All the oceanic modes have neutral growth rates. The spatial structures of some of the oceanic modes are comparable to observed spatial structures of sea surface temperature variations in the Pacific Ocean. Most years to decades variability of meridional modes of the ocean model is contained in tropical and midlatitude modes. Some oceanic modes with years to decades periods have standing oscillations in the tropics and poleward propagation of zonal velocity and layer thickness outside the tropics. Uncoupled, free atmospheric modes in the presence of mean meridional circulations have oscillation time scales ranging from a week to several decades. Such low-frequency meridional modes do not exist in the atmospere model if there are no mean meridional circulations. A large number of modes are grouped around intraseasonal time scales. Unlike the oceanic modes, the atmospheric modes are weakly unstable. Most of the intraseasonal variability of atmospheric modes is contained in tropical, midlatitude, and polar modes. Atmospheric modes with oscillation periods longer than about one year have global extent. Meridional ocean-atmospheric modes exist in the models wherever there are mean meridional circulations, i.e., tropical, midlatitude, polar, and global. Oceanic and atmospheric eigenvectors have symmetric (assymetric) latitudinal structures if their basic states are symmetric (asymmetric) around the equator. For both models, models calculated at coarser than 2.5° grid spacing do not accurately represent low-frequency variability. Scale analysis shows taht advection by tge basic state meridional velocities is the primary cause of the meridional oscillations on time scales longer than two years in the ocean model and longer than a few weeks in the atmosphere model. Meridional modes of the coupled ocean-atmosphere models are the subject of a subsequent paper.