# UK WEATHER & AIR QUALITY FORECASTING TEACHING PORTAL

## The Laws of Physics for the Atmosphere – The Computer Model

The second requirement to produce a forecast is an understanding of how one state of the atmosphere evolves to another. This evolution is encapsulated within the laws of physics as developed for the Earth's atmosphere. The following are the five required laws.

• Ideal gas law
• Conservation of mass (air)
• Conservation of mass (water)
• How wind changes
• How temperature changes

### Ideal Gas Law

The ideal gas law, also known as the equation of state, describes the relationship between the pressure ($p$), density ($\rho$), and temperature ($T$) of the air in the atmosphere. The letter $R$ is called the ideal gas constant. The ideal gas law has several different forms, and the form below is commonly used.

$$p=\rho RT$$

#### FACT BOX What is the gas constant?

The gas constant is a quantity that is universal and does not change with time. Used in the ideal gas law, it equals 8.314 J K−1 mol−1.

### Conservation of Mass (Air)

Matter can neither be created nor destroyed. As such, a rather simple equation can be written for the conservation of mass in the atmosphere. It is called the continuity equation. It describes the change in air density ($\rho$) with time as the result of the three-dimensional divergence of air.

$$\frac{1}{\rho}\frac{d\rho}{dt}+\nabla\cdot\mathbf{V}=0$$

In other words, if more air flows out of a region than enters the region (mass divergence), then the density will decrease. Alternatively, if more air flows into a region than exits the region (mass convergence), then the density will increase.

### Conservation of Mass (Water)

Water is an important chemical constituent in the Earth's atmosphere. Without the water in the atmosphere, we'd have no rain. Expressing the conservation of water mass in the atmosphere is simple.

$$\frac{dq}{dt}=Q_{E}-Q_{C}$$

In the equation above, the amount of water in a small volume of air is called the mixing ratio $q$. The mixing ratio will increase ($\frac{dq}{dt} > 0$) if water vapor is evaporated into the air ($Q_{E}$). The mixing ratio will decrease ($\frac{dq}{dt} < 0$) if water vapor is condensed out of the air ($Q_{C}$).

### Equation of Motion

Another equation essential for prediction is the momentum equation, also known as the equation of motion. The equation of motion is based on Newton's second law, which states that an acceleration (or a change in the momentum or velocity) is caused by the application of a force.

$F = ma$

$F$ = force
$m$ = mass
$a$ = acceleration

If all the forces acting in the atmosphere are included, we end up with the following for the equation of motion for the atmosphere.

$$\frac{d\mathbf{V}}{dt}+2\Omega\times\mathbf{V}=\frac{1}{\rho}\nabla p-g\mathbf{k}+v\nabla^{2}\mathbf{V}$$

The equation above says the following. The change in the three-dimensional velocity (or an acceleration, $\frac{d\mathbf{V}}{dt}$) plus the Coriolis force ($2\Omega \times \mathbf{V}$) is equal to the gradient of pressure ($\nabla p$) minus gravity ($g$) plus the friction within the fluid due to viscosity ($v\nabla^{2}\mathbf{V}$).

Gravity only works in the vertical direction and is largely offset by the vertical gradient of pressure (i.e., pressure decreases with height, creating a vertical pressure gradient force).

A similar expression can be derived for the forces acting in the horizontal direction. In that direction, the dominant forces are the horizontal pressure gradient force and the Coriolis force.

#### FACT BOX The Coriolis Effect

The Coriolis effect was discovered in the 19th Century by French mathematician Gustave Coriolis. The Coriolis effect is the apparent deflection of air in the atmosphere because the Earth rotates underneath it. An observer in space would not see the deflection, which is why the Coriolis force is referred to as an apparent force, because it needs to be accounted for within a rotating frame of reference. On Earth, the Coriolis effect affects large-scale motions, on the scale of several hours and longer because of the Earth's low rate of rotation (one revolution per day).

### Thermodynamic Equation

The last equation describes how temperature changes in the atmosphere.

$$c_{v}\rho\frac{dT}{dt}+p\nabla\cdot\mathbf{V}=Q_{H}+Q_{D}$$

The first term on the left-hand side describes the rate of change of temperature with time ($\frac{dT}{dt}$). The other term on the left-hand side says that the horizontal divergence of air ($\nabla\cdot\mathbf{V}$) leads to vertical motions that warm or cool the air. The two $Q$ terms on the right-hand side express heat fluxes from the Earth's surface to the atmosphere ($Q_{H}$; e.g., cold air moving over water water is warmed by sensible heat fluxes from the water to the air) and diabatic heating ($Q_{D}$) caused by condensation (warming the atmosphere) and evaporation (cooling the atmosphere). Calculating these terms will yield how the temperature changes with time in the atmosphere.

### Programming the model

These are the laws of physics written in a mathematical form. This mathematical form, however, is not how a computer would recognize these equations. So, the equations need to be rewritten from their mathematical form to what is called finite-difference form in order to be programmed into an algorithm (or computer code) which is used in the forecast model.

With the initial conditions derived from the global observations and the computer model derived from the laws of physics, we are ready to run the model and get a forecast.

These pages are written for ManUniCast by David Schultz, Fiona Lomas, and Katy Mulqueen, University of Manchester. Photos and graphics are credited individually.