A global climate model typically contains enough computer code to fill 18,000 pages of printed text; it will have taken hundreds of scientists many years to build and improve; and it can require a supercomputer the size of a tennis court to run. ¹

Climate modelling is similar to weather forecasting but at a timescale of decades instead of hours. Some metereological services use climate models for both.

At their most basic level, climate models use equations to represent the processes and interactions that drive the Earth’s climate. These cover the atmosphere, oceans, land and ice-covered regions of the planet.

A number of fundamental physics principals go into models:

• The first law of thermodynamics. i.e "in a closed system, energy cannot be lost or created, only changed from one form to another".
• The Stefan–Boltzmann law, i.e. that greenhouse gases naturally keep the earth about 33°C warmer than it would be.
• The Clausius-Clapeyron equation which "characterises the relationship between the temperature of the air and its maximum water vapour pressure".
• The Navier-Stokes equations of fluid motion, which "capture the speed, pressure, temperature and density of the gases in the atmosphere and the water in the ocean."

Climate models are typically written using Fortran!

The energy balance model

• Earth gets ALL energy from the sun. In turn, it returns energy to space. In general: $\texttt{energy in} = \texttt{energy out}$
• Energy from sun that hits the earth is called a "solar flux", measured as watts/m².
• Energy in can roughly be calculated as $\frac{S}{4}$, where S is a solar flux.
• $\texttt{energy out} = \texttt{radiated energy} + \texttt{reflected energy}$
• "Albedo" is the reflectivity of the earth.
• Reflected energy is calculated as $\frac{AS}{4}$, where $A$ is the albedo
• The remaining energy is absorbed, either into the atmosphere or into Earth's surface. It's then converted to infrared energy and emitted from Earth to space as infrared radiation.
• A "black body" is a perfect absorber and emitter of radiation. Earth is not a black body, but it's relatively close to being one.
• Radiated energy is calculated as $\sigma{}T^4$, where $T$ is the temperature of Earth and $\sigma{}$ is the Stefan–Boltzmann constant.
• Therefore:

$\texttt{energy in} = \texttt{energy out} = \frac{S}{4} = \frac{AS}{4} + \sigma{}T^4$