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The question is pretty simple: If I know the weather where I stand, I can estimate the weather 5 meters or 1 km away away pretty well, but I'll have a hard time guessing what the weather is, say, 50 km away.

Therefore, it seems that the climatic system has a length-scale. Where does it come from? Navier-Stokes equations do not feature an internal length scale, and it doesn't seem that the scale comes from earth's radius either.

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It seems important to recognize that there are a lot of length scales at play here. The Earth's radius or the height of the atmosphere are IR scales at play, but turbulence makes energy dissipate to small vortices (of < 1mm?) in the UV. – Vibert Feb 6 '13 at 12:31

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.

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The wave length in lakes is determined by the lake size? This sounds weird to me. The wavelength is typically a few orders of magnitude smaller. – yohBS Feb 6 '13 at 13:28
@yohBS of course it is much smaller than the size but the size does play a crucial role in building up waves. That is why in an ocean one can have a ten meter wave heights and ten meter wavelengths whereas in a swimming pool some centimeter ones and centimeter wavelengths. – anna v Feb 6 '13 at 13:34
I agree, but that's a totally different effect. It would be awefully wrong to say that the typical length-scale of the waves in lakes stems from the lake size, scaling-wise. – yohBS Feb 6 '13 at 14:27
It is not scaling wise, it is boundary condition wise, for people who have solved a differential equation. – anna v Feb 6 '13 at 14:36

Part of your scale comes from the observer, the time and space averaging of the observations.

You say the weather 10 meters distant is pretty similar to here, but look a bit closer, and it is not. The wind 10 meters from where you are can be quite different from where you stand. A sunny rock can be dry and hot 1 meter away from a shady moist spot with moss growing.

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