Sorry for being quiet for a week. I have now some time to reply to your posting.
You did say something about the size of the pan. You asserted that there are “tiny wavelets”, which I agree would exist in a small pan. If the pan is a thousand miles wide (and deep) motion would be dominated by large waves and eddies, as predicted by a large Reynolds number.
There is a characteristic length scale in the ocean. It’s the depth, which is very large but not infinite. Flow is dominated by eddies and waves, as with high Reynolds number flows.
First, again my apologies for not reading my own text. You are right of course that I implicitly referred to the size of the pan ('tiny wavelets'). But I believe it does not make a difference, see further down.
As for your second argument, I would argue depth is NOT a characteristic length scale in the ocean as far as wind-driven waves are concerned. The depth is so large that it is in fact infinite as far as these waves are concerned because they do not interact with the bottom. In other words, the waves would have identical behavior if the ocean were infinitely deep as if it were, say, 1Km deep. In other words, you could not distinguish at the surface between the two conditions.
You are of course right that there is turbulence in the wave layer but this is not captured by the standard Reynolds number which is relative to the dimension of some fixed object, like the diameter of a pipe or an air foil. I do not know enough hydrodynamics to give a precise definition what it is in the case of ocean waves but I believe it is something like the distance between the ocean surface and the boundary layer between turbulent and non-turbulent flow (perhaps the van Dorn book that you recommended addresses this; I have received it but it came just before I flew out and I am now on a different continent, so I can not consult it). This boundary layer is not a fixed object so the simple derivation of Re from the Wikipedia article does not apply. But in any case, the ocean floor can not be the 'object' used in the derivation of Re because then it would mean that the whole ocean is turbulent. Instead, turbulence occurs only in a very thin layer (on the order of 10m), everything below that is laminar.
Nevertheless, in spite of differences in the details, I think we agree in the main part, namely that there is turbulent flow at the top of the ocean. However, I believe this does not address the question of relative motion, see below.
Your continued invocation of Einsteinian relativity is an unnecessary complication. Velocities are nowhere near the speed of light, and gravity is 1G. Newtonian mechanics is sufficient to model this phenomenon, but it must include the nonlinear inertial terms, which is why the simple difference between air and water current do not fully describe the wave motion. Due to inertia, the relative direction causes a different behavior for 10 kt relative co-directional flow than 10 kt counter-directuinal flow. None of this violates relativity, or any other laws of physics.
Even Einstein’s Special Relativity requires selecting a non-accelerating reference frame, or the laws of physics are different (additional terms for fictitious forces):
Your suggestion of setting the frame of reference moving with a single molecule of water would be an accelerating frame of reference due to the chaotic motion of the water.
OK, there is more to answering this than I want to type here. You are right that you don't need Einstein's theory of relativity. As far as 'Newtonian mechanics' it depends on what you understand by it. Newton seemed to believe that there is an absolute frame of reference (the 'fixed stars' that I referred to earlier) which he thought was necessary to explain, e.g. that if you rotate a bucket of water around its long axis the water will climb up the walls. In contrast, if you imagine that you yourself accelerate around the bucket, you might get dizzy but you will not see the water rise up the walls of the bucket. His explanation was that this difference arises because the bucket moves relative to this absolute frame of reference. To make this mathematically treatable, 'fictious forces' were introduced in the presence of acceleration (not sure if Newton did this already, I am not a historian of science). Many have argued that this makes not much sense (like Ernst Mach, and even before that) but it was Einstein who made everything crystal-clear.
The laws of physics are the same in all frames of reference, accelerating or not. It is a common misconception that special relativity only applies to non-accelerating frames of reference. The first sentence in the space.com site that you refer to, "In 1905, Albert Einstein determined that the laws of physics are the same for all non-accelerating observers" is correct but misleading: the laws of physics are also the same for all ACCELERATING observers. A few lines down in that article is a sentence that is clearly wrong: "Einstein then spent 10 years trying to include acceleration in the theory and published his theory of general relativity in 1915". This is nonsense, the difference between special and general relativity is not that the first does not include accelerating frame but that the first is valid in the special case of a flat space-time, ie in the absence of gravity (or constant gravity), and the latter does include gravity.
This issue of fictional forces is explained at https://en.wikipedia.org/wiki/Inerti...e_of_reference
. I can't explain it shorter or better than they do. The most important sentence from that essay is this:
"In practical terms, the equivalence of inertial reference frames means that scientists within a box moving uniformly cannot determine their absolute velocity by any experiment. Otherwise, the differences would set up an absolute standard reference frame."
Ref. 21 is Einstein's book, ref 22 is a collection of lectures by one of my favorites, Dick Feynman (I had the pleasure of seeing him lecture in person, he was even more clear then than in the books that collect his lectures in writing). Neither is easy reading but Feynman starts at the basics and the book is available for free: https://nirstern.files.wordpress.com...asy-pieces.pdf
The sentence in red is one of the foundations of physics, just like the conservation laws of mass/energy, momentum etc. No violation of them has ever been found so we take them as gospel. Therefore, imagine you have a mass of water large enough that you can neglect the effects of its boundaries (which otherwise would establish a frame of reference) with a given current (vector) of velocity v_c. Now you blow wind over it, say with a strength v_s in the same direction as the current. The relative velocity of the wind over the water is (v_s-v_c). This will create some kind of wave pattern. Now take the same ocean with current velocity 0 and wind velocity v_s-v_c (same values as before). The wave pattern (and anything else) must be IDENTICAL in the two cases. Because, if that were not the case, we could define an absolute frame of reference, e.g. as that one that generates the first wave pattern. This is impossible. Therefore, any complications that you introduce (nonlinear interactions, fictitious forces, whatever) that you introduce can not violate any of these fundamental laws. If they do, something is wrong with these additions.