# Using mathematical equations in our computer models

All computer models of the atmosphere are based upon the same set of five governing equations, but they differ in the approximations and assumptions made in the application of these equations.

Equations of motion - the movement of an object under different circumstances

Thermodynamic equation - conservation of energy

Continuity equation - conservation of mass

Equation of state - state of matter under a given set of physical conditions

Water vapour equation - how the amount of water vapour changes depending on conditions

## Equations of motion

Newton's second Law of Motion - the law of acceleration - states *the rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the same direction. *Basically it explains how an object will change velocity if it is pushed or pulled upon by other forces.

The main forces in the atmosphere are:

the Coriolis force;

gravity;

pressure differences.

In simple terms, as air begins flowing from high to low pressure, the Earth rotates under it, making the wind follow a curved path.

In the Northern Hemisphere, the wind turns to the right of its direction of motion. In the Southern Hemisphere, it turns to the left. The Coriolis force is zero at the equator.

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. Hydrostatic equilibrium explains why the Earth's atmosphere does not collapse to a very thin layer on the ground.

Many computer 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.

## Thermodynamic equation

The First 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, as a consequence of local expansion or contraction, or from other local effects such as evaporation or condensation which are important when dealing with clouds.

## Continuity equation

Continuity equation is the basic principle of Conservation of Mass, which essentially states that matter is neither created or destroyed, although it may be rearranged. For example, rising air in one location means downward motion somewhere else; southerly flow here requires northerly flow elsewhere. These weather principles ultimately come from 'continuity'.

## Equation of state

An equation of state describes the state of matter under a given set of physical conditions:

Pressure

Density

Temperature for a perfect gas

The atmosphere obeys this equation quite well. Together with gravity, the equation of state is the key link between dynamics and thermodynamics, and enables 'heat' to drive motion.

## Water vapour equation

A water vapour equation describes the way in which the amount of water vapour in a particular parcel of air changes as a result of transportation and condensation/evaporation (where clouds are involved).

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