Follwing text is courtesy of Richard Harvey:
If Earth's sphericity is of prime importance as the ultimate cause of weather and climate, Earth's *ROTATION* is almost as important, although the atmosphere would still exhibit motions even if it wouldn't rotate. However, rotation, together with sphericity, causes a certain kind of wave motion over the mid- and high latitudes called Rossby waves, after the scientist who first described them in the 1930's.
In other words those waves occur precisely becausethe Earth is round and turns on itself. Take any one of those two conditions away,and these waves become mathematically impossible. Now, those waves are in a sense the "building blocks" of the so-called low and high pressure systems that cause much of the day-to-day variability of the weather as we experience it in our latitudes (typically between 40 deg. and 60 deg. North and South). Those waves travel generally from west to east and there are usually three to five along a latitude circle. Sometimes a wave can be "blocked" by an uptream one and cause the associated weather patterns to litterally stagnate over an area for many days or even weeks, until the blocking wave erodes itself and starts to move eastward again.
So we see that, apart from the fact that it is the sun's energy which is the primal ingredient, the fundamental causes of our weather and climate on this Earth (and for many other planets in our solar system for that matter) are of a GEOMETRIC nature.
The Coriolis force is the force directly caused by Earth's rotation, and should be mentioned when going into a more detailed description of the physics of Rossby waves. In the Northern Hemisphere, this force will cause horizontally moving objects to curve to the right, while in the Southern Hemisphere the opposite is true. In the context of the large-scale motions of the atmosphere (and the Rossby wave), the Coriolis force combines with the horizontal pressure force to produce a subtle dynamic balance which maintains a West to East wind at pretty much all heights (again in the mid-latitudes only!), in BOTH Northern and Southern Hemispheres. When the north-to-south temperature difference becomes too high, we say that this West-to-East flow becomes unstable and wants to "break down" into wavy patterns whose motions will tend to redistribute the heat in a more equitable manner, thus transporting cold and dry polar air equatorward, and warm and moist tropical air poleward. Those waves are the Rossby waves, upon which your average "weather system" with clouds and rain and/or snow will be superimposed.
Now, what I've said so far is only half the story for the following reason: we see from the above that without Earth's rotation and the associated Coriolis force, the balance between it and the pressure force wouldn't exist, and thus one cannot maintain the west-to-east flow upon which Rossby waves grow and decay. It is important to mention on the other hand that without Earth's rotation, the atmosphere would still transport heat poleward but through a different regime called the Hadley Cell regime. This is presisely the way the atmosphere responds in an area where the Coriolis force is very weak or even zero: the tropics! Why is it weak there? That's because what is important is the "vertical" component of the the Earth's rotation which produces the Coriolis force, and the Coriolis force is greatest when the local vertical is parallel to the Earth's rotation axis, that is, the poles, and it vanishes at the equator where the local vertical is perpendicular to it. So really the Earth's rotation isn't "felt" equally by the atmosphere everywhere, but only where the Coriolis force is sufficiently high, that is, far enough away from the equator. In our case, this roughly happens poleward of 30 deg. north/south.
The Hadley Cell regime is the most direct way for a fluid to transport heat: from the heated ground below, this induces rising motion at the equator, then poleward motion aloft (tranporting warm air) and equatorward motion near the ground (transporting cold air). It is similar in many ways to a kettle boiling water. Poleward of 30 deg. north/south, this regime breaks down and the growing Coriolis force produces the Rossby wave regime which we discussed earlier.
The existence of those two regimes of the atmposphere is the reason why I placed rotation "not quite" as important as the equator-to-pole differential heating from the sun as the ultimate cause of atmospheric motions. This primal differential heating produces two different responses of the atmosphere, depending on whether the local air "feels" the Earth rotating underneath.
All this because of the unique properties of a fluid flowing over a rotating sphere!
Well, the wavelength of a Rossby wave depends primarily on the vertical stability of the atmosphere, and yes on the rotation rate of the earth. From theory, it turns out that a faster rotation would likely produce (all other things being equal) more waves around a latitude circle. On the other hand, fluid viscosity plays for all intents and purposes no role on the large-scale motion of the atmosphere, only when one considers motions of sub-centimeter scale, where molecular viscosity can be felt. To go into more detail would require more time than what I have at the moment! Sorry!