## Setting up a numerical atmospheric model

A numerical model of the atmosphere contains descriptions, in appropriate computer form and with necessary approximations, of the basic dynamics and physics of the different components of the atmosphere and their interactions.2 When a physical process is described in terms of an algorithm (a process of step-by-step calculation) and simple parameters (the quantities that are included in a mathematical equation), the process is said to have been parameterised. The dynamical equations are:

• The horizontal momentum equations (Newton's Second Law of Motion). In these, the horizontal acceleration of a volume of air is balanced by the horizontal pressure gradient and the friction. Because the Earth is rotating, this acceleration includes the Coriolis acceleration. The 'friction' in the model mainly arises from motions smaller than the grid spacing, which have to be parameterised.

• The hydrostatic equation. The pressure at a point is given by the mass of the atmosphere above that point. Vertical accelerations are neglected.

• The continuity equation. This ensures conservation of mass.

The model's physics consists of:

• The equation of state. This connects the quantities of pressure, volume and temperature for the atmosphere.

• The thermodynamic equation (the law of conservation of energy).

• Parameterisation of moist processes (such as evaporation, condensation, formation and dispersal of clouds).

• Parameterisation of absorption, emission and reflection of solar radiation and of thermal radiation.

• Parameterisation of convective processes.

• Parameterisation of exchange of momentum (in other words, friction), heat and water vapour at the surface.

Most of the equations in the model are differential equations, which means they describe the way in which quantities such as pressure and wind velocity change with time and with location. If the rate of change of a quantity such as wind velocity and its value at a given time are known, then its value at a later time can be calculated. Constant repetition of this procedure is called integration. Integration of the equations is the process whereby new values of all necessary quantities are calculated at later times, providing the model's predictive powers.

Since computer models for weather forecasting were first introduced, their forecast skill has improved to an extent beyond any envisaged by those involved in the development of the early models. As improvements have been made in the model formulation, in the accuracy or coverage of the data

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