Opening

One of the aspects about our Math for Sustainability course that I love is the ability to tailor the mathematics to the appropriate level for the students without sacrificing the lessons it can teach us about sustainability and Earth systems. I want us to explore the idea of Tipping Points. Upon preparing myself to teach this topic, I experienced probably my first moment where I could clearly notice an unintuitive lesson springing from the mathematics in the course.

Motivation

One unintuitive aspect about climate change is the nonlinear responses to changing conditions. In fact, these changes can be even be non-smooth. It’s intuitive to imagine over time the temperature of the Earth ticking up slowly, growing ever warmer in a smooth way. But scientists would quickly tell you that that’s not always how it works. Let’s perform our own derivation of Earth’s global temperature imagined as a Stock (of heat in the atmosphere) with Inflows and Outflows (under a continuous family of models for Outflow).

Derivation

1. Inflow, Outflow Curves

Inflow

We assume the only relevant inflow of energy to the Earth is directly from the Sun. We model that energy flow using Stefan’s Law [the amount of thermal radiation emitted by each square meter of the surface of an idealized physical object at temperature $T$ Kelvin is $\sigma T^4$, where $\sigma$ is called Stefan’s constant and is approximately equal to $5.7 \times 10^{-8} \text{ W} / \text{m}^2 \cdot \text{K}^4$] The Sun is modeled by a sphere of radius $R_{\text{Sun}} =7 \times 10^8 \text{ m}$ with surface temperature $T_{\text{Sun}} = 5,800 \text{ K}$. In total, the Sun then outputs

$4 \pi R_{\text{Sun}}^2 \cdot \sigma T_{\text{Sun}}^4 \approx 4 \times 10^{26} \text{ W}$.

To translate this to the Earth’s incoming energy, we take note that

• Not all energy received by the Earth is absorbed: denote by $\alpha$ the Earth’s albedo, or the proportion of energy received by that gets reflected off the surface. Earth’s current albedo is close to $\alpha \approx 0.3$ (mostly cloud cover, some from deserts & ice & snow).
• Earth is at a distance from the Sun and receives only a proportion of its total output: we calculate the solar constant $L$, or the amount of solar radiation that would fall on a $1 \text{ m}^2$ panel directly/perpendicularly facing the Sun. Earth’s orbital radius is $d = 1.5 \times 10^{11} \text{ m}$, so $L = \frac{4 \times 10^{26} \text{ W}}{4 \pi \cdot d^2} \approx 1,350 \text{ W}/\text{m}^2$.

Call the radius of Earth $R_{\text{Earth}}$.

$\text{Inflow } = (1 - \alpha) \cdot\pi R_{\text{Earth}}^2 \cdot L$

Imagining the Earth as a sphere without an atmosphere, its outflow would follow Stefan’s Law:

$\text{Outflow } = 4 \pi R_{\text{Earth}}^2 \cdot \sigma T_{\text{Earth}}^4$

We can sophisticate our outflow model by imagining there is indeed a greenhouse gas effect caused by having an atmosphere. Let us make a simple model and identify a greenhouse gas parameter $g$, affecting the Outflow as follows:

$\text{Outflow } = \frac{4 \pi R_{\text{Earth}}^2 \cdot \sigma T_{\text{Earth}}^4}{1 + g}$

We specially designate the model at $g = 1$ to be the Glass Ball Model, characterized as having equal amounts of black body radiation coming from the Earth either being trapped by the atmosphere or escaping to space; whereas $g = 0$ is called the Naked Planet Model.