Exercise 200 — Simple climate model¶
A minimal climate–carbon system¶
📓 Notebook
simple_climate_model.ipynb
Learning objectives¶
By completing this exercise, you will learn to:
- Translate CO₂ emissions into changes in atmospheric concentration
- Compute radiative forcing from CO₂ using a logarithmic law
- Simulate the temperature response using a zero-dimensional energy balance model
- Understand the role of system inertia, feedbacks, and time scales
- Interpret numerical results in a policy- and engineering-relevant way
This exercise is a thinking tool, not a predictive climate model.
Model overview¶
The notebook implements two related models of increasing physical realism.
Model A — Temperature anomaly + CO₂ concentration¶
State variables:
- \(T(t)\): global-mean temperature anomaly (°C)
- \(C_{\mathrm{atm}}(t)\): atmospheric CO₂ (ppm)
Temperature equation¶
\[
\frac{dT}{dt}
=
\frac{
F_0 + 5.35\ln\!\left(C_{\mathrm{atm}}/280\right)
-\lambda_{\mathrm{cl}}\,T
}{C}
\]
- \(C\): effective heat capacity (response time)
- \(\lambda_{\mathrm{cl}}\): climate feedback parameter
- \(F_0\): background forcing term (optional)
Carbon cycle (one-box)¶
\[
\frac{dC_{\mathrm{atm}}}{dt}
=
E_{\mathrm{ppm}}(t)
-\beta\left(C_{\mathrm{atm}}-280\right)
\]
- Emissions are converted using
\(1\,\mathrm{ppm} \approx 2.13\,\mathrm{GtC}\) - \(\beta\) represents net carbon uptake by sinks
Model B — Blackbody energy balance + carbon mass¶
This variant tracks absolute temperature \(T\) (K) and carbon mass (GtC):
\[
C\,\frac{dT}{dt}
=
\frac{S(1-\alpha)}{4}
+
5.35\ln\!\left(\frac{C_{\mathrm{atm}}}{C_{\mathrm{atm},0}}\right)
-
\sigma T^4
\]
It highlights:
- the nonlinear nature of outgoing radiation
- the difference between temperature anomaly and absolute temperature
How to run the notebook¶
jupyter lab
Guided questions¶
Use these questions to structure your exploration.
1) Emissions vs temperature¶
- What happens to temperature if emissions become zero instantly?
- Does temperature stop increasing immediately?
- Why or why not?
2) Time scales¶
- Which variable responds fastest: emissions, CO₂ concentration, or temperature?
- Which parameter controls this behavior?
3) Carbon sinks¶
- How does changing the uptake parameter \(\beta\) affect long-term CO₂?
- Is temperature recovery symmetric with temperature increase?
4) Feedback strength¶
- How does \(\lambda_{cl}\) influence equilibrium temperature?
- Can you interpret it physically?