How to calculate the upper limit on the number of days weather can be forecast reliably? To put it bluntly, weather is described by the Navier-Stokes equation, which in turn exhibits turbulence, so eventually predictions will become unreliable.
I am interested in a derivation of the time-scale where weather predictions become unreliable. Let us call this the critical time-scale for weather on Earth.
We could estimate this time-scale if we knew some critical length and velocity scales. Since weather basically lives on an $S^2$ with radius of the Earth we seem to have a natural candidate for the critical length scale.
So I assume that the relevant length scale is the radius of the Earth (about 6400km) and the relevant velocity scale some typical speed of wind (say, 25m/s, but frankly, I am taking this number out of, well, thin air). Then I get a typical time-scale of $2.6\cdot 10^5$s, which is roughly three days.
The result three days seems not completely unreasonable, but I would like to see an actual derivation.
Does anyone know how to obtain a more accurate and reliable estimate of the critical time-scale for weather on Earth?
 A: I don't think that such a computation of a theoretical limit of accuracy is possible. There are several sources of uncertainty in weather models:


*

*initial and boundary data,

*parameterizations,

*numerical instability, rounding and approximation errors of the numerical scheme employed to solve the Navier-Stokes equations for the atmosphere.
The term "parameterization" refers to the approximation of all subgrid processes, these are all processes/influences that happen at a scale that is smaller than the length of a grid cell. This includes effects from the topography, or the local albedo. More sophisticated approximations to subgrid processes can actually lead to less predictivity of a model, because the needed more detailed initial and boundary data are not available. 
The Navier-Stokes equations themselves are usually approximated up to a minimum length scale that is way larger than the length scales that would be necessary to resolve turbulent flows, these kinds of approximations are called large eddy simulations.
The accuracy of this truncation depends critically on the kind of flow and turbulence.
While I don't think that it is possible to derive a theoretical limit, what people do instead is performing ensemble runs, where the results of a weather model are compared that are calculated with slightly perturbed initial and boundary data.
An example of such a "twin" experiment can be found here:


*

*David Strauss, Dan Paolino: Intermediate time error growth and predictability: tropics versus mid-latitudes
(The result is that the error becomes significant after ca. 15 days of simulated time.)
A: I am not sure how useful this "back of the envelope" calculation of reliability of Numerical Weather Prediction is going to be. Several of the assumptions in the question are not correct, and there are other factors to consider.
Here are some correcting points:


*

*The Weather is 3 dimensional and resides on the surface of the planet up to a height of at least 10km. Furthermore the density decreases exponentially upwards. Many atmospheric phenomena involve the third dimension such as rising and falling air circulation effects; jet streams (7-16km).

*The equations are fluid dynamics plus thermodynamics. The Navier-Stokes equations are not only too complex to solve, but in a sense inappropriate as well for the larger scales. One problem is that they might introduce "high frequency" effects (akin to every individual gust of wind or lapping of waves), which should be ignored. The earliest weather prediction models were seriously wrong because the high frequency fluctuations of pressure needed to be averaged rather than directly extrapolated. Here is a possible equation for one point of the atmosphere:
Tchange/time = solar + IR(input) + IR(output) + conduction + convection + evaporation + condensation + advection
The regionality of the model is important too. In a global model there will be larger grid sizes and sources of error from initial conditions and surface and atmosphere top boundary conditions. In a mesoscopic prediction there will be smaller grid sizes but sources of error from the input edges as well. The smallest scale predictions of airflow around buildings and so on might be a true CFD problem using the Navier-Stokes equations however.
I dont know that any calculation is done to predict the inaccuracies, although the different types including the numerical analysis (chaos) error sources can be studied separately. Models are tested against historical data for accuracy overall with predictions made 6-10 days out.
To assume that the atmosphere "goes turbulent" after 3 days seems to conflate several issues together.
A: First, you would need to define a "reliable" forecast, e.g. x% confident of temperature, wind, rain,etc within some limits.
Second, your question presumes that weather is not more or less deterministic.  At some point, human behavior produces effects that influences the chaotic model.  However, the essential problem with chaotic models is the sensitivity to the initial conditions.  This results in limitations due to data collection as well as computational issues, e.g. rounding errors.  It is not clear what are the inherent limits to model improvement.
