[jerf]

In addition to the other fine replies, weather forecasting's inaccuracies are dominated by the lack of information, then by lack of processing power. Lack of closed-form solutions to NS or better solutions rates quite a ways down the list of issues it has, or put another way, even if we had a magic box that completely accurately solved NS for weather forecasting, it would not get that much accurate. (My suspicion is that it would literally be measured in "minutes" more accurate, rather than the "days" you'd like, but I concede I can't prove that... but bear in mind that it may well be the cases that it would be milliseconds more accurate or something, not just that I could be wrong about it being "days", as the errors in the initial data compound over the course of the simulation no matter what math you throw at the problem.)

[btrettel]

It seems popular to believe that for fluids in general forecast accuracy is dominated by errors in the initial conditions (ICs), but my own look at the problem suggests that's not so clear. I recall skimming a book on forecasting by a weather forecaster and he addressed this misconception. It appears that there are multiple sources of error, from errors in the ICs to numerical integration to the fact that the models they use are approximate (i.e., they don't solve NS; they solve a filtered version of NS with a turbulence model and additional models for other physics like chemistry), etc. My impression is that the dominant two are model inadequacy (the models are approximate) and compounded errors due to IC errors and non-linearity, but which is larger likely depends on the problem, and I am not particularly confident about this in general as I don't have hard data. (Certain types of turbulence models get more accurate as the resolution/computational cost increases, but I can't speak for other models. This fits with what you said about lack of computational power.)

The right way to do this is through uncertainty quantification techniques, and I don't know a lot about those at the moment. Until then, all I can say is that there are multiple sources of error.