Refer to a report here that mentions the huge advancement in weather forecasting models since the 1980s:
How much better? “A modern five-day forecast is as accurate as a one-day forecast was in 1980,” says a new paper, published last week in the journal Science. “Useful forecasts now reach nine to 10 days into the future.”
But weather forecasting cannot be indefinitely accurate, one fundamental limitation is Chaos Theory:
Chaos: When the present determines the future, but the approximate present does not approximately determine the future. (Wikipedia)
I used to think some ten or twenty years ago that the accuracy level of 3 day weather forecasts was due to Chaos Theory, but apparently it wasn't.
Are there studies where weather forecasting models are subjected to sensitivities of initial conditions, investigating how Chaos Theory puts an upper limit on the accuracies of such models? Also, based on current models, how much further can weather forecasting models go before the limits of Chaos Theory set in?