How Does a Meteorological Model Work ?
Accuracy and Resolution
Dr. John W. (Jack) Glendening, Meteorologist 
Updated:  Dec 31, 2007

This webpage gives a basic description of how meteorological models generate their predictions, with an emphasis on model errors and the influence of model resolution.  It is intended to provide a framework for discussions of weather prediction models and their errors.  Additional information on errors of particular importance to BLIPMAP predictions is given in the Model Notes webpage.

The Grid

      Numerical meteorological models produce predictions by solving equations at grid points created by dividing the atmosphere into cells and forecasting the average of each predicted variable over that cell.  Generally cells are regularly spaced in the horizontal but vary in the vertical, with vertical spacings becoming smaller near the surface since atmospheric conditions change more rapidly with height there.  Similarly, vertical cell spacings in general are much smaller than horizontal spacings since conditions change much more rapidly in the vertical than in the horizontal.  (See the Grid Cell Diagram

The Equations of Motion

      A meterological model produces forecasts by solving the "equations of motion" for a fluid, which are derived from fundamental physical laws: conservation of mass (both of air and moisture), conservation of momentum (Newton's laws), and conservation of thermal energy (thermodynamics). These equations predict the change in variables such as temperature or velocity resulting from the physical phenomena which affect them. These equations strongly interact with each other - for example, a change in thermal energy (affecting temperature) will change atmospheric density (affecting mass), in turn changing pressure differences (affecting velocity). These feedback interactions tend to counter a forced change, e.g. to oppose an atmospheric heating by introducing a cooling effect. If forcing were to remain constant then eventually the feedbacks would produce an "equilibrium" in which changes with time would become increasingly smaller - but the forcing of the atmosphere is never constant, as solar heating is constantly changing, so the atmosphere is in a state of constantly adjusting "quasi-equilibrium".

Solution Methodology

      What is being solved is a set of "Partial Differential Equations", where the PDEs predict the time-change of a variable based upon conditions in each central cell and its horizontal/vertical neighbors.  Those who have taken calculus will remember that differential equations only apply to "infinitesimally" small spacings so we are assuming that a "finite difference" over the finite grid spacing is a reasonable approximation to a differential.  The equations are solved in a "marching-forward-in-time" fashion, starting from the assumed 3D initial conditions (which are usually based on available observations at that time), and predicting how the variables change at each time step at each grid point to give forecasts at a given time.  PDEs are solved for the "prognostic variables" of temperature, humidity, the three components of velocity, condensed moisture (clouds), etc.  The model also calculates "diagnostic" variables which depend solely upon conditions at the present time (so they depend upon the prognostic variables but do not have to be solved in a "time-marching" fashion).  Since all the equations are inter-related, an error in any one will to some degree affect the others.  These PDEs are not empirical equations, since they are fundamentally derived from well-established physical laws of conservation of energy and momentum and mass.

Model Initialization

      Since the equations of motion predict changes there must be an initially prescribed 3D atmosphere for them to start from. This is obtained from atmospheric observations (baloon, satellite, and aircraft data) at a limited number of points which are then interpolated to complete the 3D grid. There will of course be errors in the interpolated initial state (it is not a solution of the equations of motion), so initially the equations predict relatively large changes as they evolve towards the equation of motion quasi-equilibrium. Forecasts during this "spinup" period are subject to large errors and should be taken with a grain of salt, since they reflect the initial "interpolated" guess. Spinup inaccuracies decrease with time - within the BL, the minimum-to-maximum spinup time is roughly 1-to-4 hours, the longer time being for larger grid spacings and larger initial errors (spinup times are longer above the BL).

Overall Prediction Accuracy

      Forecast accuracy of the many parameters predicted by a meteorological model can be generally ordered, from most accurate to least accurate, as:  (1) Winds,  (2) Thermal parameters,  (3) Moisture parameters,  (4) Cloud parameters,  (5) Rainfall. 

Model Errors

      The accuracy of the a model depends on many factors, which can be roughly grouped as:

Accuracy and Model Resolution

      Many of the errors listed above are reduced when the spatial resolution of a model is increased, notably those errors resulting from surface averaging effects, from lack of resolved topography, and from the model's finite-difference equations lacking proper resolution of actual atmospheric differences.  Therefore meteorologists are constantly striving to obtain more powerful computers so that they can use ever smaller grid spacings (in both the horizontal and vertical) to reduce these errors and provide better predictions.  Of course, there will continue to be errors due to other factors, such as initial condition errors, which cannot be reduced by using a finer resolution - but it is best to at least eliminate the errors that one can control.  BL predictions are particularly affected by grid resolution errors, and often much less affected by other errors, so better model resolution is especially valuable for soaring predictions.  Some parameterization errors are reduced as resolution is improved, but other are not so parameterziation error will often continue to be a significant source of BL prediction error, particularly for cloud forecasts.
      I should point out that the computer power needed for improved model resolution increases as the fourth power of the resolution increase, e.g. halving the grid spacing requires the computer power to be 16 times larger!  So doubling a model's resolution is a big accomplishment!   [This is because in addition to a factor of 8 increase corresponding to the increased number of horizontal and vertical grid points, since the finer resolution should occur in all directions, there is an additional factor of two because the time step must now be halved to accommodate the smaller grid size.]