ernestyalumni reblogged your photo and added:
Great answer. I learned a lot. To avoid, for myself and others, your initial drove you crazy, could you reference some good or great books, papers, pdf notes that explain these dimensionless quantities clearly and/or rigorously?
Thanks.
There aren't really any ideal resources on this kind of thing (IMO), though Barenblatt's books "Scaling" and "Scaling, Self-similarity, and Intermediate Asymptotics" may be worth looking at for a different perspective from the ones you've encountered -- he initially treats nondimensional quantities only as tools for constructing self-similar exact solutions, and only later looks at how the behavior of these exact solutions may relate to how the PDE behaves more broadly (when an exact solution can't be written).
What stopped my "going crazy" phase was not reading any particular text, but reading and doing enough fluid mechanics that I got used to the idea that these numbers were often very useful for building both intuition and useful reduced equations, and that one rarely has to worry about precisely how they are defined, because they tend to work unproblematically if they work at all. (If the Reynolds number of your flow is large enough then it is probably pretty large no matter how you play with ambiguities in the definition of the Reynolds number; if the ambiguities do matter, then that the usual asymptotic approach based on Reynolds number is too crude and you should do something else.)
So maybe an example of how these numbers build intuition will be most useful. Here is a handy example, more relevant to immediate experience than turbulence stuff. Why do cyclones rotate, and why do they rotate in different directions in the northern and southern hemispheres?
As you may know, cyclones center around regions of unusually low pressure. From basic fluid intuitions, you'd expect the air to flow down the pressure gradient toward the center. Instead, it moves steadily in circles around the center. And the direction of the circle is counter-clockwise in the northern hemisphere and clockwise in the southern. What gives?
Well, it turns out that if you write F = ma for a little parcel of fluid in the cyclone, you get something like
mass*acceleration = (Coriolis force) + (pressure gradient)
The Coriolis force is a fictitious force produced by the earth's rotation. It depends on a parameter, "f," which is positive north of the equator and negative south of it. And it depends on the velocity (!) -- specifically it is always perpendicular to it.
Now, it turns out that if you look at the typical net forces felt by a parcel in a cyclone, they're much smaller in magnitude than either of the terms on the right side above. That is, the Coriolis force and the pressure gradient approximately cancel, producing a sum much smaller than either of them individually. So it is approximately true that:
0 = (Coriolis force) + (pressure gradient)
(This does not mean that the acceleration is zero! Just that the error in the most recent equation is much smaller than either of the terms in it.)
But since the Coriolis force depends on the velocity, we can determine the velocity from the above directly, without having to integrate acceleration in time. Specifically, since the Coriolis force is perpendicular to the velocity, it looks like f x v (a cross product between a set vector f and the velocity v) and we have
-f x v = pressure gradient
From this we can get v, and it is perpendicular to the pressure gradient. And its direction will depend on the direction of f, explaining the flipped behavior in the two hemispheres. (This is called "geostrophic balance" -- a "balance" in this context means two or more forces approximately cancel.)
What does this all have to do with nondimensional numbers? Well, the easiest way to tell if something is closest to geostrophic balance is to postulate some typical scales of motion and plug those scales into the F = ma equation. The acceleration term will be smaller than each of the forces by a factor equal to the Rossby number.
Now, do we have to postulate scales? What exactly do we mean when we say a motion has a "typical length scale" L and a "typical velocity scale" U? It's a subtle question; our cyclone presumably has lots of turbulent fluctuations at small space/time scales that would make it pointless to define the Rossby number at every point (x,y,t) as though it were meaningful. So there's some sort of low-pass filtering* going on here -- we're selecting large scales as the ones of interest. It is a very robust, if not a priori obvious, result that this kind of thing usually works -- saying "I'm interested in the large-scale behavior of this cyclone" and postulating scales based on its large-scale motion tends to give you a good model, and more generally, saying things like "my flow has low Reynolds number so it can be well-described as a Stokes flow" tend to work in spite of the difficulties involved in precisely defining the Reynolds number, so long as any sensible definition would give you a small answer. (If the answer is truly small, technicalities shouldn't matter -- that's asymptotics for you.)
(I think this is why there is so little interest in coming up with a precise definition of what these numbers "really mean" -- a wealth of experience, much of it in observational contexts with limited data, confirms that this kind of broad and vague thinking works surprisingly well, and trying to nail down "the Rossby number means this and this only" would inevitably rule out the useful work of some subset of previous researchers on some technicality or other.)
Anyway, my point here is that the only way one can intuitively understand cyclones is by understanding geostrophic balance. And the Rossby number quantifies how geostrophically balanced a system is likely to be. This generally works, and a more precise (i.e. well-defined) diagnostic for geostrophic balance is not only unnecessary but perhaps not desirable; it would involve some formal criteria that might be unsatisfiable in an observational context, even for flows that anyone with their intuitions screwed on right would say is "low Rossby number."
*or statistical ensemble averaging, and yes there's a difference. But that's a matter for another day.
