Figure 1. A very simple STELLA model of Earth’s climate system.  The three colored sectors show the parts of the model that keep track of the energy coming in to the Earth from the Sun, the energy leaving the Earth through emitted heat, and the average surface temperature of the Earth.

Click for a text description of the STELLA Diagram for the Planetary Climate Model.

This STELLA diagram visually represents the planetary climate model by categorizing key climate-related variables into three main groups: Energy In (yellow area), Temperature (red area), and Energy Out (blue area).

• Energy In (Yellow Area):
  • Factors affecting incoming solar energy: Solar Constant, Albedo, Surf Area.
  • These variables influence Insolation, which directs energy to Earth Heat.
• Temperature (Red Area):
  • Includes Ocean Depth, Water Density, and Heat Capacity, which influence Temperature.
  • Temperature interacts with Earth Heat and contributes to heat exchange processes.
• Energy Out (Blue Area):
  • Outgoing energy factors include Heat Emitted, LW Int, and LW Slope.
  • Earth Heat emits energy, balancing the system.

Arrows show the flow of energy and interactions between these factors. The diagram helps illustrate the components regulating planetary climate.

David Bice @ Penn State is licensed under CC-BY-NC-4.0

The Energy In sector (yellow in Fig. 1 above) controls the amount of insolation absorbed by the planet.  The Solar Constant is not really a constant, but it does tend to stay close to a value of 343 Watts/m² (think of about six 60 Watt light bulbs shining down on a patch of ground 1 meter on a side — this is what we get from the Sun). This is then multiplied by (1 – albedo) and then the surface area of the Earth giving a result in Watts (which is a measure of energy flow and is equal to Joules per second). In the form of an equation, this is:

$${\mathrm{E} }_{\mathrm{in} }= \mathrm{S} \times \mathrm{A} \times ( 1 - \alpha )$$

S is the Solar Constant (343 W/m²), A is surface area, and α is the albedo (0.3 for Earth as a whole).

This is the equation ${\mathrm{E} }_{\mathrm{in} }= \mathrm{S} \times \mathrm{A} \times ( 1 - \alpha )$

Figure 1. A very simple STELLA model of Earth’s climate system. The three colored sectors show the parts of the model that keep track of the energy coming in to the Earth from the Sun, the energy leaving the Earth through emitted heat, and the average surface temperature of the Earth.

Click for a text description of Figure 1.

This STELLA diagram visually represents the planetary climate model by categorizing key climate-related variables into three main groups: Energy In (yellow area), Temperature (red area), and Energy Out (blue area).

• Energy In (Yellow Area):
  • Factors affecting incoming solar energy: Solar Constant, Albedo, Surf Area.
  • These variables influence Insolation, which directs energy to Earth Heat.
• Temperature (Red Area):
  • Includes Ocean Depth, Water Density, and Heat Capacity, which influence Temperature.
  • Temperature interacts with Earth Heat and contributes to heat exchange processes.
• Energy Out (Blue Area):
  • Outgoing energy factors include Heat Emitted, LW Int, and LW Slope.
  • Earth Heat emits energy, balancing the system.

Arrows show the flow of energy and interactions between these factors. The diagram helps illustrate the components regulating planetary climate.

David Bice @ Penn State is licensed under CC-BY-NC-4.0

The Energy Out sector (blue above) of the model controls the amount of energy emitted by the Earth in the form of infrared (thermal) radiation, which is a form of electromagnetic radiation with a wavelength longer than visible light, but shorter than microwaves. You saw earlier that this is often described using the Stefan-Boltzmann Law which says that the energy emitted is equal to the surface area times the emissivity times the Stefan-Boltzmann constant times the temperature raised to the fourth power:

A is the whole surface area of the Earth (units are m²), ε is the emissivity (a number between 0 and 1 with no units), σ is the Stefan-Boltzmann constant (units are W/m² per °K4), and T is the temperature of the Earth (in °K). The problem with this approach is that it ignores the greenhouse effect, which is a very important part of our climate system. We could represent the greenhouse effect by choosing the right value for the emissivity in the Stefan-Boltzman law, but here, we will use a different approach, one in which Eout is based on actual observations. With a satellite above the atmosphere, we can measure the amount of energy emitted in different places on Earth and figure out how it relates to the surface temperature. As it turns out, this is a pretty simple relationship, described by a line:

$${\mathrm{E}}_{\text{out}}= ( ⟦ \mathrm{LW} {⟧}_{\mathrm{int}}+ ⟦ \mathrm{LW} {⟧}_s\times T ) \times A$$

The part inside the parentheses is just the equation for a line, with an intercept (LW_int with units of W/m²) and a slope (LWs with units of W/m² per °C). This new way of describing Eout is shown as the red line in the figure below:

Figure 2. Three different schemes for representing the long-wavelength energy (heat) emitted by Earth. The blue curve is the simple Stefan-Boltzman Law, which suggests that at the average temperature of the Earth (15°C), our planet would emit way more energy than we get from the Sun, and so we would cool down until the temperature reached -18°C at which point the Ein = Eout and we have a steady state. The green curve shows the Stefan-Boltzman Law modified by including a new term called emissivity (0.6147), which brings us into an energy balance (steady state) at a temperature of 15°C. The red curve instead represents this relationship based on actual measurements from satellites — notice that it too puts us at a steady state when the temperature is 15°C. The red curve is what we will use in this model.

Click for a text description of Figure 2.

This graph plots Long-wave Energy Emitted (W/m²) on the y-axis against Temperature (°C) on the x-axis to illustrate how Earth's natural greenhouse effect raises the planet’s temperature.

• The blue curve represents E_out = σT⁴, showing energy emitted without an atmospheric greenhouse effect.
• The green line represents E_out = 0.6147σT⁴, depicting energy emission with Earth's greenhouse effect.
• The red line represents E_out = 210 + 2T, another approximation of outgoing energy.
• A horizontal dashed line at 240 W/m² represents energy absorbed from the Sun (343(1 - albedo)).
• The intersection of the green line with the 240 W/m² level indicates Earth's actual average surface temperature of 15°C.
• Without the greenhouse effect, the temperature would be -18°C, marked on the graph.
• A label notes that Earth’s greenhouse raises the temperature by 33°C.

David Bice @ Penn State is licensed under CC-BY-NC-4.0

The key thing here is that the hotter something is, the more energy it gives off, which tends to cool it, and it will continue to cool until the energy it gives off is equal to the energy it receives — this represents a negative feedback mechanism that tends to lead to a steady temperature, where ${\mathrm{E} }_{\text{in }}= {\mathrm{E} }_{\text{out }}$.

Figure 1. A very simple STELLA model of Earth’s climate system. The three colored sectors show the parts of the model that keep track of the energy coming in to the Earth from the Sun, the energy leaving the Earth through emitted heat, and the average surface temperature of the Earth.

Click for a text description of Figure 1.

This STELLA diagram visually represents the planetary climate model by categorizing key climate-related variables into three main groups: Energy In (yellow area), Temperature (red area), and Energy Out (blue area).

• Energy In (Yellow Area):
  • Factors affecting incoming solar energy: Solar Constant, Albedo, Surf Area.
  • These variables influence Insolation, which directs energy to Earth Heat.
• Temperature (Red Area):
  • Includes Ocean Depth, Water Density, and Heat Capacity, which influence Temperature.
  • Temperature interacts with Earth Heat and contributes to heat exchange processes.
• Energy Out (Blue Area):
  • Outgoing energy factors include Heat Emitted, LW Int, and LW Slope.
  • Earth Heat emits energy, balancing the system.

Arrows show the flow of energy and interactions between these factors. The diagram helps illustrate the components regulating planetary climate.

David Bice@ Penn State is licensed by CC-BY-NC-4.0

The Temperature sector (brown in Fig. 1) 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: (see directions for how view images in a larger format)

$$T = \frac{E}{A \times d \times \rho \times C_{p }}$$ or $$\text{Temperature }= \frac{\text{Energy }}{\text{area} \times \text{depth} \times \text{density }\times \text{heat capacity }}$$

Let’s look at it with just the units, to make sure that things cancel out:

$$\left[ { }^{\mathrm{O} } \mathrm{K} \right] = \frac{[ \mathrm{J} ] }{\left[ {\mathrm{m} }^{2 }\right] \times [ \mathrm{m} ] \times [ \mathrm{kg} ] \times \left[ {\mathrm{m} }^{- 3 }\right] \times [ \mathrm{J} ] \times \left[ {\mathrm{kg} }^{- 1 }\right] \times \left[ { }^{\circ }{\mathrm{K} }^{- 1 }\right] }$$

This can be simplified by combining, rearranging, and canceling to give:

$$[°K]=\frac{\cancel{[m3]}\times \cancel{[kg]}\times \cancel{[J]}\times [°K]}{\cancel{[m3]}\times \cancel{[kg]}\times \cancel{[J]}}$$

Here, E is the thermal energy stored in Earth’s surface [Joules], A is the surface area of the Earth [m²], d is the depth of the oceans involved in short-term climate change [m], ρ is the density of seawater [kg/m³] and Cp 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. It also includes some other features that allow you to change the solar input and the part of the greenhouse effect due to CO₂. We use the standard assumption (which is itself based on some physics calculations) that for each doubling of the CO₂ concentration, there is an increase of 4 W/m² in the greenhouse effect. This is often called the greenhouse forcing due to CO₂. In terms of our E _out curve shown in Figure 2 above, this shifts the red curve downwards — so less energy is emitted, and thus more is retained by the Earth. Let’s consider how this works — if we start with 200 ppm of CO₂ and increase it to 800 ppm, that represents 2 doublings (from 200 to 400 and then from 400 to 800), so we would get 8 W/m² of greenhouse forcing.

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 0.1 years.