Medium-scale motions are defined here as motions whose horizontal space scales L are much smaller than the Earth's radius r0 but still large enough that they are not directly affected by molecular diffusion. They are the middle component of our triple decomposition into large-, medium-, and small-scale motions that arises from two Reynolds decompositions. The first decomposition separates large- from medium- and small-scale motions. The second decomposition separates large- and medium-scale motions from small-scale motions. Large-scale motions must thus parametrize the eddy fluxes caused by medium- and small-scale motions. Mediumscale motions must prescribe the large-scale fields and parametrize the eddy fluxes caused by small-scale motions. Small-scale motions must prescribe the large- and medium-scale motions. Diffusion is molecular. Medium-scale motions thus face two closure problems, one with respect to larger scale motions and one with respect to smaller scale motions. The only simplification is one of geometry. The smallness of the parameter γ := L/r0 allows the spherical geometry to be approximated by a variety of “planar” geometries. These approximations are only valid locally and include:

• midlatitude beta-plane approximation;

• equatorial beta-plane approximation;

• f-plane approximation; and

• polar plane approximation.

These are geometric approximations that are similar to the spherical approximation that relied on the smallness of Earth's eccentricity d20 /r20 and of the parameter H/r0, and led to the introduction of pseudo-spherical coordinates.