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All numerical models of the atmosphere are ultimately based upon the same set of governing equations which are discussed here in non-mathematical terms. Numerical models differ in the approximations and assumptions made in the application of these equations.
Equations of Motion
Newton's 2nd Law of Motion states that the acceleration of a particle is equal to the vector sum of forces acting upon that particle. It is a statement of the Conservation of Momentum principle. The main forces in the atmosphere are gravity, the Coriolis force and the force that acts on air due to differences in pressure. The Coriolis force accounts for the apparent acceleration that air possesses by virtue of the Earth's rotation. In the absence of other forces, the path of a moving air parcel, relative to the surface of the Earth, will not be straight but will be curved. The curve will be towards the right in the northern hemisphere and to the left in the southern hemisphere. The Coriolis force allows for this effect.
In the horizontal the pressure difference and Coriolis force are the main causes of acceleration.
In the vertical the two main forces are gravity and the pressure gradient due to the variation of pressure with height. In fact, the gravitational force is almost exactly balanced by the pressure gradient force, a condition known as hydrostatic equilibrium. Many models assume hydrostatic equilibrium, but our model does not. This means it can take account of strong vertical wind motion, making it suitable for running at very high resolution.
The vertical component of the Coriolis force is also included in our model. Although it is comparable in magnitude with the horizontal components, it is negligible when compared against the gravitational and vertical pressure gradient forces separately. For this reason it is often ignored, but it can be significant in regions of strong vertical motion, and is retained for completeness and self-consistency.
The 1st Law of Thermodynamics requires that the amount of heat added to a system is exactly balanced by the work done in increasing its volume and the increase in internal energy. It is an expression of the principle of the conservation of energy.
Temperature at a point in the atmosphere can change either due to cooler or warmer air being blown to that point, or as a consequence of local expansion or contraction, or from other local effects such as evaporation or condensation.
This is the basic principle of Conservation of Mass which essentially states that matter is neither created or destroyed.
Equation of State
The equation of state relates the three primary thermodynamic variables, pressure, density and temperature for a perfect gas. The atmosphere is assumed to obey this equation. Though no real gas is perfect, and the atmosphere is, additionally, a mixture of gases, the perfect gas assumption is a very good approximation.
Water Vapour Equation
This describes the way in which the amount of water vapour in a particular parcel of air changes as a result of advection and of condensation/evaporation.
Progress in numerical weather prediction techniques is regularly reported in a number of scientific journals such as the Quarterly Journal of the Royal Meteorological Society and Monthly Weather Review.
There are a few text books available which provide more detail on the equations described above.