As per wikipedia, scientists have not been successful to accurately predict the whether which is 2 weeks ahead. Here is the excerpt:

The atmosphere is a chaotic system, as a result, small changes to one part of the system can grow to have large effects on the system as a whole.This makes it difficult to accurately predict weather more than a few days in advance, though weather forecasters are continually working to extend this limit through the scientific study of weather, meteorology. It is theoretically impossible to make useful day-to-day predictions more than about two weeks ahead, imposing an upper limit to potential for improved prediction skill

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    $\begingroup$ One of the initial explorations into chaos theory happened because a scientist (forget his name, sorry) was running a weather simulation, and would get different results from the same initial parameters. It turned out that when he was saving/reloading the numbers, the lowest bit of the floats was getting rounded. The effect magnified every processing cycle, so the simulations started out the same, but eventually completely diverged. $\endgroup$ – Ask About Monica Jul 7 '17 at 23:40
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    $\begingroup$ @kbelder I do not believe this for a second - floating point maths is not random, this is a common misconception. It is entirely deterministic. If you put in the same numbers, you get the same result, unless your algos have some sort of randomness. If they do not, then either the results are the same, or your algorithm is misbehaving. That being said, very small changes in inputs resulting in very different outputs is exactly chaos theory, and is related to bifurcation theory. $\endgroup$ – will Jul 8 '17 at 10:31
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    $\begingroup$ @will yes, yes and yes! what can happen, though, with FP arithmetic, is that save/restart (checkpointing), or two runs of the same models compiled with different compilers on different systems may produce very different results (given a long enough time integration window), an effect that can be entirely attributed to the difference in computer architectures and FP math implementations. This will become more and more important as people move to exascale where hardware faults are expected to occur much more often $\endgroup$ – GoHokies Jul 8 '17 at 11:02
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    $\begingroup$ ... anyway, at present, the uncertainties in the models (physics, boundary conditions, sub-grid parameterizations, etc.) and observations have imho much larger effects on forecast quality than these FP math "perturbations" $\endgroup$ – GoHokies Jul 8 '17 at 11:02
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    $\begingroup$ @will: If I understand kbelder correclty, he is saying that the scientist got different when running the simulation directly, vs running after a save + reload which introduced an additional rounding. Consider also the x87, which can operate internally using 80 bits, while using less when going to/from memory. $\endgroup$ – ninjalj Jul 8 '17 at 11:03

Further to Chris' answer:

Yes, weather (or the equations describing it) is extremely sensitive to the initial conditions. The fact that the weather system contains phenomena at pretty much all time and space scales (have a look at figures 1.1 and 1.2 of these notes for some examples) does not make predictions any easier. Also, the "weather" (at various scales) is dependent on a whole bunch of other planetary systems (oceans, land-based processes, etc.) that come in with their own model and observational uncertainties:

Figure 2 of https://www.nature.com/nature/journal/v525/n7567/full/nature14956.html

Figure taken from Bauer et al., Nature, 2015 showing "physical processes of importance to weather prediction".

That said, significant progress has been made in the past few decades, and people in the operational weather forecasting community (at least the folks over at ECMWF, which are really at the top of their game when it comes to weather forecasting) refer to this graph that showcases these improvements:

A measure of forecast skill at three-, five-, seven- and ten-day ranges, computed over the extra-tropical northern and southern hemispheres.

The figure above is from Bauer et al., Nature, 2015 and shows a measure of forecast skill at three-, five-, seven- and ten-day ranges, computed over the extra-tropical northern (NH) and southern hemispheres (their caption).

What the figure shows is that forecast skill has been steadily increasing for the past 3 decades, with some major "bumps" in performance brought in by

  • the advent of satellite (space-borne) measurements, which significantly narrowed the gap between the southern and northern hemispheres (the ground-based/sea observational network in the SH is much sparser than that of its northern counterpart).
  • the introduction of a probabilistic (ensemble-based) weather forecasting framework

For example, the 10-day forecast quality is the same as that of a 7-day forecast made 30 years ago, so you can say we're up a day every 10 years :)

So I, for one, think that there is no hard physical barrier (boo, Lorenz!) that does not/will not allow us to predict the weather, say, 14 or 20 or 25 days in advance. It's just that we're lacking (1) sufficient knowledge of the weather system (2) computing power to simulate the weather at high-enough resolution (3) high-quality observational data to constrain existing 3D/4D-var and ensemble forecasting systems.

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    $\begingroup$ Does the last paragraph of your answer imply that, to the best of your knowledge, it is NOT "theoretically impossible" to make long-term predictions, but is instead "practically impossible" or "theoretically improbable"? (my own intuition: it does seem rather impractical to make long-term predictions about chaotic systems, but "theoretically impossible" is, afaik, a higher bar) $\endgroup$ – Soron Jul 8 '17 at 16:20
  • $\begingroup$ i don't see why it would be "theoretically impossible" (that's what i meant by a hard physical barrier). is there anything truly random (truly unpredictable) about the atmospheric state, i.e., what we call 'weather'? i, for one, don't think so. however, barring a large leap in knowledge of the atmospheric dynamics / computing / observational data coverage , it may well remain practically impossible in the foreseeable future. $\endgroup$ – GoHokies Jul 8 '17 at 19:14

Errors grow exponentially in a chaotic system, and most people believe weather is chaotic. So even if you get a fairly exact numerical approximation, the fact that your input data (temperatures, etc.) and your model are slightly off from reality will cause results which exponentially diverge from reality as time goes on. Thus long time prediction of weather is extremely difficult!

  • $\begingroup$ In terms of people expressing an opinion, I'd go as far as saying most people believe that weather is chaotic, with those believing in deterministic forcing allowing long term forecasts being the outliers. $\endgroup$ – origimbo Jul 7 '17 at 15:47
  • $\begingroup$ So your answer implies that computers that can minimize the "exponential growth of errors" in chaotic systems are non-existent in terms of technology. If yes, then why? $\endgroup$ – Muhammad Maqsoodur Rehman Jul 7 '17 at 15:59
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    $\begingroup$ @MuhammadMaqsoodurRehman The problem isn't simply at the level of computation, but in the underlying physics. The system is so nonlinear that trajectories which start off close together diverge in time. This means you need to know the precise state of the system, as well as how the boundary conditions will change in the future (down to the level of whether the proverbial butterfly will flap its wings) to make a long term prediction. $\endgroup$ – origimbo Jul 7 '17 at 16:27
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    $\begingroup$ @MuhammadMaqsoodurRehman The butterfly might be an exaggeration, but effects on the scale of the behaviour of a single thundercloud are usually agreed to affect weather as experienced on a human scale. $\endgroup$ – origimbo Jul 7 '17 at 18:37
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    $\begingroup$ Seems like no-one is mentioning that the problem of predicting inherent weather system complexity (the chaotic nonlinear dynamics) is hugely compounded by having extremely coarse-grained input data (in both time and space). Such poorly approximate inputs are not very good at constraining even a good short term prediction (i.e. next 6 hours) for small-sized locales. Hence the need to make loose predictions ("20% chance of rain today") based on averages from ensemble models. $\endgroup$ – RHC Jul 12 '17 at 0:02

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