Separate response time from equilibrium

Separate the destination from the pace

Compare effective depths 50 and 200 m while holding the feedback coefficient at 1.2 W m⁻² K⁻¹. Both receive the same imposed forcing and have the same radiative response at a given temperature anomaly. Predict whether they must nevertheless follow different temperature trajectories.

The deeper effective reservoir stores more energy per degree, so it adjusts more slowly. If the 4 W/m² forcing were maintained indefinitely, both cases would approach the same equilibrium anomaly. The actual experiment removes forcing after 25 years, so that sustained-forcing equilibrium is a conditional target, not a guaranteed endpoint.

A simple energy balance links storage with radiative response. The distinction between forcing, response and energy accumulation also appears in IPCC AR6 Chapter 7. The coefficients and reservoir here are teaching choices, not estimated climate-system parameters.

Find the conditional equilibrium

Write λ = 3.2 − f for net damping, in W m⁻² K⁻¹. Across the controls λ remains positive. During the imposed forcing, net flux is N = 4 − λθ. At equilibrium N = 0, so the conditional target is θeq = 4/λ.

At f = 1.2, λ = 2 and θeq = 2 K. Heat capacity does not appear in this equilibrium equation. It controls how quickly energy changes temperature, rather than the temperature at which the specified incoming and outgoing changes balance.

On paper, compare H = 50 m and H = 200 m. Write the same target beside both, then their different capacities: 205 and 820 million J m⁻² K⁻¹. This separates a balance condition from a storage property before the curve adds visual detail.

Interpret the response time

The response time is τ = C/λ, initially in seconds. Divide by 31,557,600 to express it in model years. At the default depth and feedback, τ is about 3.248 years. At 200 m with the same feedback, τ is four times larger, about 12.992 years.

While forcing remains active, θ(t) = θeq[1 − exp(−t/τ)]. The exponential starts at one, making θ initially zero, then decreases toward zero. After one response time, the temperature has completed about 63.2% of the approach to its target; it has not completed the adjustment. After three response times, about 95% is complete.

You can reason with these fractions without deriving the exponential. Draw a target line and mark 63.2% and 95% of its height. Predict where each depth’s trajectory lies at the same elapsed time. The interface keeps time fixed across cases so the comparison concerns response rather than mismatched observation dates.

Inspect an incomplete adjustment

At year 25, the 50 m, f = 1.2 case is about 1.9991 K above its initial state. The 200 m case is about 1.7080 K. Both have a 2 K sustained-forcing target, but the larger reservoir has not approached it as closely before removal.

After removal, the target becomes zero and each trajectory decays toward it. The larger reservoir can remain warmer later even though it warmed less during the forcing interval. This is a consequence of storage and response time, not a reversal of energy conservation.

Capture both H/f recipes and record initial warming rate, τ, conditional target and year-25 temperature. Explain why no single one of these quantities completely describes the response. A rapid initial rate is not itself a large equilibrium sensitivity.

The model has one reservoir and constant coefficients. It cannot represent multiple ocean adjustment times, geographic variation or exchange between a surface layer and deep ocean. Its effective depth is not an inferred real mixing depth.

Transfer by changing capacity

A stable model has a response time of four years and a sustained-forcing target of 3 K. Its heat capacity is tripled with forcing and damping unchanged. Predict the new response time, target and initial warming-rate ratio.

Check the response-time transfer

The response time becomes twelve years, the target remains 3 K, and the initial warming rate becomes one third as large. Capacity changes the pace of adjustment, not the equilibrium balance under these fixed assumptions.