A start at answering the question:
Forecasting the weather requires;
1.a measurement of the initial conditions and
2. mathematical model which can then be used to project forward in time the initial conditions
A system called numerical weather prediction (NWP) forms the basis of modern weather forecasting.
The system uses a mathematical model of the atmosphere which has been derived from the laws of physics.
This model provides a set of equations to solve in order to predict the future weather. These equations are solved by averaging over 'chunks' of the atmosphere (grid boxes) and short periods of time (time steps) to then give us numerical equations which are put into the supercomputer.
The size of a chunk is called the 'resolution' of the model, similar to the resolution of an image from a digital camera.
More than one grid box is needed to represent weather features in the much the same way as a digital image needs more than one pixel to represent something like a face. The number of equations which have to be solved depends on the total number of grid boxes.
As the equations have to be solved long before the weather happens, the size of grid boxes (given the forecast area) is limited by how quickly they can be solved.
The current global forecast model has a horizontal resolution of about 40 km over the UK, meaning 160 million equations have to be solved just to step the atmosphere 15 minutes in time. This resolution is very good for information about the general weather conditions over the UK, and the Met Office's current computer power means a 5-day forecast can be produced in a few hours.
At shorter range (1-2 days) a higher-resolution model (about 12.5 km) is used, because it provides more regional detail.
Dynamics Research develops and maintains the dynamical core of the Met Office's Unified Model and undertakes research on numerical techniques for use in future cores.
• About Dynamics Research
Dynamics Research is responsible for all aspects of the dynamical core of the Met Office Unified Model.
• Even Newer Dynamics for General atmospheric modelling of the environment (ENDGame)
ENDGame is the current operational dynamical core for the Unified Model and is based on a semi-implicit semi-Lagrangian discretization of the governing equations.
• GungHo - a next generation atmospheric dynamical core for weather and climate modelling
To be able to run effectively on the next generation of supercomputers, future atmospheric dynamical cores will need to scale on hundreds of thousands of processors.
• Continuous equation sets and approximations.
The partial differential equations governing atmospheric flow are the subject of this topic.
• Conservative semi-Lagrangian transport schemes
SLICE (Semi-Lagrangian Inherently Conserving and Efficient) is a semi-Lagrangian transport scheme that inherently conserves mass.
• Physics-Dynamics Coupling
The tendencies due to the physical parametrization schemes need to be coupled to those of the dynamical core.
• Atmospheric dynamics
• Atmospheric physics
• Atmospheric composition
• Quantifying forecast uncertainty
• Forecast evaluation
Atmospheric physics is a vital part of a weather forecast model and is often referred to as the physical parametrization. ECMWF research focuses on how to represent unresolved physical processes in the atmosphere, such as radiation, clouds and subgrid turbulent motions.
The reference to unresolved processes points to a major issue in limiting forecasts; not all of the phsical processes are resolvedand hence models are far from perfect.
Another limiting factor is measuring the initial state of the atmosphere; and remembering the fast number of iterative steps that are necessary to produce forecasts, small initial errors increse and incfease. Thtais why one technique currently beibg used is run ensembles of forecasts. The initial conditions are varied slightly , maybe 50 times and the model is then run 50 times and the results are compared. Usually the forecasts will gradually tend to diverge at some time but the longer they stay close together the greater the confidence out to that time.
All of this of course requires enormous computing power and so therein lies another limitatio;the amount of computing power available.
There is a vast amount of info on weather forecasting, developmeent and research but the few links Ive included, to UKMO nad ECMWF will provide anyone with a vastsource of material. All it takes is time to read and follow all of the links.
The attached image covers the processes and interactions that feed into producing our weather and so have to be modelled and as of yet not all of the processes are fully understood or modelled mathematically..