Immediate? Certainly not. Weather models do have incomplete information, but the equations are approximated by Taylor series (1st to 3rd order, depending on the model, last time I checked).
It's possible that this can explain some of the differences between models or ensemble runs... but you have to realize that most of the error comes from incomplete data in the initial and boundary conditions. Looking for weather model effects is like looking for relativistic effects in automobiles.
Convergence, or at least consistency and order, of Runge-Kutta methods is shown via Taylor series, so I don't see the problem or why the GP is only "Almost" correct.
Geez guys, I chose a term that I thought most people would be more likely to understand. I realize that some of us have worked a lot more on numerical methods, but other hackers never made it past Calculus 2.
Did you take offense from my comment? All I wanted is to set people on the right track if they want to find more information about how NWP is implemented.
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.)
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.
Fluid dynamicist here. At the risk of looking like a heretic, I am willing to say that I don't think the NS Millennium Prize problem and related things will end up being very useful practically.
I can't see how a definitive answer to the question will result in better turbulence modeling, which is what matters from a practical point of view. If it turns out that the solutions are not unique then we could probably find an additional condition to add (e.g., the entropy condition) to make the solutions unique. If the solutions are unique, bounded, etc. then that's great and it would have no impact practically speaking aside from perhaps helping the reputation NS has for accuracy. Some people seem to think that solving the NS Millennium Prize problem would likely lead to a solution for the turbulence problem, but as I said, I can't see how. I'd be interested if anyone could explain this belief better.
There may be other benefits. I've found papers that find bounds on different fluid dynamics quantities to be interesting, and the motivation for these studies are the NS problem from what I understand. Unfortunately the results from these papers tend to be less useful than bounds I can derive specifically for applications myself.
(In a nutshell the turbulence problem is that NS has far too high a computational cost/complexity to be used in practical simulations. So cheaper approximations to NS are used, which you can cladsify as "turbulence models". How steep the drop-off in accuracy is as you reduce complexity is an open question. My opinion is that fluids probably require high computational cost for accuracy a-priori. Things like correlations from experiments can get around this as you are using pre-computed results, and that may be what we should go for in my philosophy.)
Financial forecasting and risk-elimination in a system like a blockchain would be my guess — and in that respect it’s probably the most important problem out there IMO.
Ha... I guess I can be a bit terse. Just think of the “turbulence” in navier-stokes as valuation fluctuations. Without the realworld dampening effects of regulation, slow tranactions, and managed markets — volatility along the lines of the infinite incongruities posed in the article are possible. ( note: Im not referring to a pendactic ‘infinite’ wrt a blockchain).
I’m an applied mathematician and a macroeconomist that studied turbulence in financial markets and crashes thereof. I can assure you that the dynamics are pretty distinct. In economics wealth is not a conserved quantity whereas in physics energy and momentum are.
I think bitcoin’s fixed-limit and deterministic transactions makes it a uniquely closed & conserved system (regardless of deflation and lost wallets). The discrepancies in valuation among exchanges & localities seem to show a relativistic (to borrow a physics term) quality. [I’m however not really into bitcoin or an economist though]
It's possible that this can explain some of the differences between models or ensemble runs... but you have to realize that most of the error comes from incomplete data in the initial and boundary conditions. Looking for weather model effects is like looking for relativistic effects in automobiles.