Trace a temperature-dependent feedback loop

Make the response return to the original change

In earlier investigations, moving an albedo or emissivity slider selected another prescribed state. Here the feedback term changes automatically as temperature changes. Temperature anomaly θ produces a radiative contribution fθ, which changes net flux, which changes the subsequent temperature. That return path is the implemented feedback loop.

Draw three boxes on paper: temperature anomaly, feedback flux and temperature-change rate. Connect them in a loop, then add the external forcing as a separate input. This helps distinguish the coefficient you prescribe from the feedback flux that responds to the model’s evolving state.

The interpretation of radiative feedback depends on sign convention. IPCC AR6 Chapter 7 discusses feedback and forcing definitions. This lab uses positive f for an additional downward-positive contribution during warming, while net damping λ = 3.2 − f is defined positive for stability. Do not transfer coefficient signs between conventions without checking their equations.

Calculate a state-dependent contribution

At the default f = 1.2 W m⁻² K⁻¹, a temperature anomaly of 1 K contributes +1.2 W/m² through feedback. At 2 K, the contribution is +2.4 W/m². The coefficient is constant, but the flux is not.

The baseline response contributes −3.2θ W/m². At θ = 1 K with forcing active, net flux is 4 − 3.2 + 1.2 = 2 W/m². At θ = 2 K, it is 4 − 6.4 + 2.4 = 0. These calculations explain why a reinforcing contribution can coexist with a stable equilibrium.

Write the three terms in separate columns for θ = 0, 1 and 2 K. Do not merge them until the final net column. A positive feedback term does not mean every contribution is positive or that net energy accumulation must continue indefinitely.

Compare reinforcing and opposing feedback

For f greater than zero, the feedback reinforces a positive temperature anomaly by reducing total damping. For f less than zero, it adds an opposing radiative response. At f = 0, the added feedback vanishes, but the baseline temperature-dependent radiative damping remains. “No added feedback” therefore does not mean “no temperature response of radiation.”

The controls keep f between −1 and +2. Net damping consequently stays between 4.2 and 1.2 W m⁻² K⁻¹. Every selectable case is stable in this linear model. None reaches zero or negative net damping, so the lab does not demonstrate a runaway or tipping point.

Compare the selected trajectory with the simultaneous f = 0 reference at the same depth. Both initially warm at F/C because θ starts at zero and all temperature-dependent terms initially vanish. Their curves diverge as θ grows. This is a useful test of whether you have confused initial forcing with feedback.

Trace sensitivity and response time together

At fixed depth, increasing f reduces λ. The sustained-forcing target 4/λ rises and the response time C/λ increases. At H = 50 m, f = 0 gives a target of 1.25 K and response time about 2.030 years. At f = 1.2, they become 2 K and about 3.248 years.

Predict these directions before comparing numbers. A stronger reinforcing feedback produces a larger eventual response to sustained forcing, but also a longer characteristic adjustment time in this model. The finite 25-year forcing interval must still be distinguished from an indefinitely maintained forcing.

The feedback is a linear radiative parameterization, not an explicit simulation of clouds, melting ice or water vapor. Its temperature response is implemented, but the physical processes that might produce such a coefficient are not resolved. It supplies no actual equilibrium climate sensitivity estimate or CO₂ calibration.

Transfer to a new coefficient

For f = −0.8 W m⁻² K⁻¹, calculate net damping and the target under maintained 4 W/m² forcing. Is the feedback reinforcing or opposing a positive anomaly? Would it change the initial warming rate at fixed capacity?

Check the feedback-loop transfer

Net damping is 3.2 − (−0.8) = 4 W m⁻² K⁻¹, giving a 1 K target. The feedback opposes a positive anomaly. Initial warming remains F/C because the feedback flux is zero at the initial θ = 0 state.