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Time-averages are common observables in analysis of experimental data andnumerical simulations of physical systems. We will investigate, from the angleof partial differential equation analysis, some oscillatory geophysical fluiddynamics in three dimensions: Navier-Stokes equations in a fast rotating,spherical shell, and Magnetohydrodynamics subject to strong Coriolis andLorentz forces. Upon averaging their oscillatory solutions in time, interestingpatterns such as zonal flows can emerge. More rigorously, we will prove that,when the restoring forces are strong enough, time-averaged solutions stay closeto the null spaces of the wave operators, whereas the solutions themselves canbe arbitrarily far away from these subspaces.

Time-averages of Fast Oscillatory Systems in Three-dimensional Geophysical Fluid Dynamics and Electromagnetic Effects