Therefore, it seems that the climatic system has a length-scale. Where does it come from?
Let us not forget that the weather system is a classical case of chaotic dynamics: several interacting differential equations are at work, not just Navier Stokes. Think of tides, think of seasons, think of clouds/albedo etc.
But mainly it is the boundary conditions to the solutions of the equations imposed by geography: mountains lakes rivers valleys seas etc. which will define a type of length. In a sense similar to the way the size of a lake defines the wavelength of the waves and the height possible : this is due to the boundary conditions they impose to the solutions of the complicated system of coupled differential equations.
Weather programs do a good job of simulating terrain and are fairly successful for a few days projections. Local projections are not always as good since systems can move faster or slower higher or lower on the map than what is programed. But the geography is taken into account.