Connect energy imbalance to temperature change

Give the energy imbalance somewhere to go

An imbalance of 4 W/m² means four joules enter storage each second per square meter at the initial state. It does not mean temperature increases by four degrees each year. To connect energy with temperature, first specify how much energy the modeled reservoir stores for each degree of warming.

Imagine two containers with the same exposed area but different amounts of water. Under the same initial energy supply per area, the larger amount requires more energy for the same temperature change. This comparison motivates heat capacity; it is not a physical experiment performed by this interface.

The lab uses an effective water-equivalent depth H, from 10 to 200 m. Its areal heat capacity is C = 4.1 × 10⁶H J m⁻² K⁻¹. The word effective matters: this is a synthetic uniform reservoir, not a measured ocean layer or a model of ocean currents. NASA describes energy exchanges within the broader Earth system in its energy-budget explanation.

Calculate an initial rate with units

At H = 50 m, C = 205,000,000 J m⁻² K⁻¹. Temperature anomaly θ begins at zero, so the initial temperature-dependent radiative terms are zero. Net incoming flux N is then 4 W/m², and the initial warming rate is N/C.

Dividing gives about 1.9512 × 10⁻⁸ K/s. Multiplying by 31,557,600 seconds per model year gives about 0.61576 K/year. This is the instantaneous initial rate, not a promise that every subsequent year warms by that amount. As θ changes, the radiative response changes the net flux.

Make a paper ledger for depths 50 and 100 m. Double the heat capacity, then predict the initial rate before inspecting the curves. The 100 m case has half the initial rate under the same forcing. Keep “joules stored per kelvin” separate from “joules arriving per second.”

Define the temperature and storage anomalies

Temperature anomaly θ is measured relative to this model’s initial equilibrium, not the 1951–1980 reference of the recorded climate dataset. A difference of one kelvin has the same size as one degree Celsius, but no absolute surface temperature is assigned here.

Stored-energy anomaly is E = Cθ. At H = 50 m and θ = 1 K, E = 205 million J/m², or 205 MJ/m². The display uses megajoules per square meter so the large numbers remain readable. One megajoule is one million joules.

The energy equation is C dθ/dt = N. Read dθ/dt as the rate at which the temperature anomaly changes. Multiplying by heat capacity converts that temperature rate back into an energy-transfer rate per area. No calculus calculation is required to interpret the units or the sign.

Follow the changing net flux

The imposed forcing is 4 W/m² for the first 25 years and zero afterward. Baseline radiative damping contributes −3.2θ W/m². A separate feedback contributes fθ W/m². Thus N = F − 3.2θ + fθ, where F is the imposed forcing and f is the selected feedback coefficient.

At the default f = 1.2, the combined temperature response is −2θ. As θ increases, this response reduces the initial imbalance. The model computes a continuous response using an exact solution, not a succession of years each receiving the initial net flux.

Inspect the ledger at years 0, 5 and 10. Reconstruct N from its three terms, then use its sign to predict whether storage is increasing or decreasing. The ledger contains synthetic calculated states, not measured ocean heat-content observations.

Transfer to an unfamiliar reservoir

A paper reservoir has C = 100 million J m⁻² K⁻¹ and an initial net input of 2 W/m². Calculate its initial warming rate in K/year and its stored-energy anomaly at θ = 0.5 K. Use the model’s seconds-per-year convention.

Check the energy-storage transfer

The initial rate is 2/100,000,000 K/s, or 0.631152 K/year. Stored energy at 0.5 K is 50 million J/m² = 50 MJ/m². The initial rate need not persist when the net flux responds to temperature.