### Introduction to a Simple Planetary Climate Model

Our first model is slightly more complicated than the diagram shown above because there are quite a few other parameters that determine how much energy is received and emitted and how the temperature of the Earth relates to the amount of thermal energy stored. The complete model is shown below, with three different sectors of the model highlighted in color:

The * Energy In* sector (yellow above - albedo, solar constant, surf area, and insolation) controls the amount of insolation absorbed by the planet. The Solar Constant converter is a constant, as the name suggests — 1370 Watts/m2. This is then multiplied by the cross-sectional area of the Earth — this is the area that faces the Sun — giving a result in Watts (which you should recall is a measure of energy flow and is equal to Joules per second). This is then multiplied by (1 – albedo) to give the total amount of energy absorbed by our planet. In the form of an equation, this is:

S is the Solar Constant (1370 W/m^{2}), A_{x} is the cross-sectional area, and a is the albedo (0.3 for Earth as a whole).

The * Energy Out* sector (blue above - surf area, LW int, LW slope) of the model controls the amount of energy emitted by the Earth in the form of infrared radiation. This is simply described by the

*as being the surface area times the emissivity times the Stefan-Boltzmann constant times the temperature raised to the fourth power:*

**Stefan-Boltzmann Law**A is the whole surface area of the Earth, e is the emissivity, s is the Stefan-Boltzmann constant, and T is the temperature of the Earth.

The * Temperature *sector (brown above - water density, ocean depth, heat capacity, temp) of the model establishes the temperature of the Earth’s surface based on the amount of thermal energy stored in the Earth’s surface. In order to figure out the temperature of something given the amount of thermal energy contained in that object, we have to divide that thermal energy by the product of the mass of the object times the heat capacity of the object. Here is how it looks in the form of an equation (with units added):

Here, E is the thermal energy stored in Earth’s surface [Joules], A is the area of the planet [m^{2}], d is the depth of the oceans involved in short-term climate change [m], ρ is the density of sea water [kg/m^{3}] and C_{p} is the heat capacity of water [Joules/kg°K]. We assume water to be the main material absorbing, storing, and giving off energy in the climate system since most of Earth’s surface is covered by the oceans. The terms in the denominator of the above fraction will all remain constant during the model’s run through time — they are set at the beginning of the model and can be altered from one run to the next. This means that the only reason the temperature changes is because the energy stored changes.

The model has a few other parts to it, including the initial temperature of the Earth, which determines how much thermal energy is stored in the Earth at the beginning of the model run. There are also some converters that divide the energy received and the energy emitted by the surface area of the Earth to give a measure of the intensity of energy flow, of the flux, in terms of Watts/m^{2}, which is a common form for expressing energy flows in climate science.

One unit of time in this model is equal to a year, but the program will actually calculate the energy flows and the temperature every 0.01 years.

Now that you have seen how the model is constructed, let’s explore it by doing some experiments. Here is the link to the model.

### The First Run

What will happen to the temperature of the Earth if we run the model for 30 years with the following initial conditions:

Initial Temp = 0°C

Albedo = 0.3 (this will not change over time)

Emissivity = 1.0 (this will not change over time)

Ocean Depth = 100 m (this will not change over time)

Solar Constant = 1370 W/m^{2}

*These are the values you see when you first launch the model.*