How feedback, memory, and delay organize recurring climate variability
ENSO is not a perfect clock. This Exploration asks how related states can return without fixed periodic timing, and how feedback, ocean memory, phase, and noise shape what can be predicted.
Change oscillator parameters and diagnose the regime
Watch time series and phase-plane motion together
Find one temperature with two different futures
Turn T(t) and H(t) into simple two-dimensional fields
Add stochastic forcing and compare multiple realizations
The oscillator evolves continuously. Its surface-temperature amplitude shapes the eastern Pacific, while its hidden heat amplitude shapes the western Pacific subsurface field.
See how a small model can isolate feedback, memory, and the mechanism behind a recurring climate rhythm.
Watch video ↗Applied Mathematics in Geosciences · Episode 6
Then explore it yourself ↓
01
Concept overview
A rhythm needs both push and memory
The recharge–discharge picture separates the fast growth of a surface anomaly from the slower ocean state that eventually changes its direction.
01 · Push
Amplifying feedback
A warm eastern Pacific can weaken the , reducing the processes that normally cool the surface and allowing warming to strengthen.
02 · Store
Subsurface memory
Heat stored below the surface evolves more slowly. It records part of the event’s history even when a surface map looks unchanged.
03 · Restore
Delayed reversal
The warm event redistributes and discharges the reservoir. The slower restoring influence can then oppose the surface anomaly.
04 · Locate
Phase is a joint state
A point in the combines the visible temperature T with the hidden heat state H, revealing where the system is in its evolution.
The same surface temperature can occur once while the event is developing and again while it is declining, because the hidden subsurface state is different.
02
Interactive Lab A
Feedback plus memory can organize a rhythm
Change the four processes in the model, then follow the same evolving state in a time-series view and a phase-plane view.
The trajectory circles while gradually shrinking. Feedback and memory create the rhythm; damping slowly removes its amplitude.
Recurrence is generated internally here: no seasonal or external clock appears in the equations.
03
Interactive Lab B
One temperature, two phases
Choose one observed surface temperature. The model finds where the same oscillation crosses it once while warming and once while cooling.
Only T is visible. At the selected level, A and B look identical even though the system is moving through different phases.
Left · time-series view of T(t)
Right · surface-only observation
A · warmingdTdtA=aT* + bHA= ?
B · coolingdTdtB=aT* + bHB= ?
A · warming
T = 0.20
HA = hidden; the surface is moving toward warmer values.
B · cooling
T = 0.20
HB = hidden; the surface is moving toward cooler values.
The surface looks the same. The hidden ocean state does not.
Knowing T alone does not determine where the system is in the cycle. H distinguishes a developing event from a declining event.
04
Interactive Lab C
How east and west take turns
The two state variables do not peak together. Their phase difference creates an alternating east–west sequence.
Sustained conceptual cycle
For this visualization we remove amplitude decay so that the phase relationship remains visible over many cycles. The purpose is to isolate the east–west sequence, not to reproduce realistic ENSO amplitude evolution.
TE(t) = A sin(ωt)·HW(t) = A cos(ωt)
Constant amplitude A = 1 · period = 24 model months · quarter-cycle phase difference
SST′(x, y, t) = TE(t) ϕE(x) ψ(y)
Subsurface′(x, y, t) = HW(t) ϕW(x) ψ(y)
See the spatial basis functions
ϕE(x) = sin[π(x + 1)4]
ϕW(x) = sin[π(1 − x)4]
ψ(y) = exp[−y22σy2]
The smooth zonal bases place their maxima in the eastern and western Pacific. The equatorial envelope uses σy = 0.42. Coordinates are normalized rather than geographic.
1 · Amplitude time series
2A · Eastern-Pacific SST anomaly
Eastern intensity TE(t) = 0.00; the color fades smoothly westward and away from the equator.
2B · Western-Pacific subsurface heat
Western intensity HW(t) = 1.00; the color fades smoothly eastward and away from the equator.
negative anomalynear zeropositive anomaly
05
Interactive Lab D
When noise turns a rhythm into an imperfect rhythm
Rapidly varying forcing changes event timing and amplitude. A nonlinear saturation term prevents large anomalies from growing without bound.
dTdt=aT + bH − βT3 + σξ(t)
ξ(t) represents rapidly varying forcing; σ sets its amplitude. The cubic βT³ is weak for small anomalies and stronger for large ones.
Time-series view
Phase-plane view
Timing changes
Successive warm events no longer peak at identical intervals.
Amplitude changes
Different realizations travel around related regions by different paths.
Structure can remain
Recurrence may still be visible even when no trajectory repeats exactly.
This is one conceptual route from a clean oscillator to an imperfect climate rhythm. Real ENSO irregularity also reflects seasonal forcing, spatial processes, changing feedbacks, and interactions with other climate modes.
06
Mathematical structure
Key mathematical ideas
Three short calculations explain why the hidden variable changes the future and when the model circles rather than returning directly.
1. Coupled first-order model
dTdt=aT + bH;dHdt=−cT − dH
T can reinforce itself through aT. H influences the surface through bH. A warm T discharges H through −cT, while −dH relaxes the memory state.
2. Same T, different tendency
(dTdt)H=H₁−(dTdt)H=H₂=b(H₁ − H₂)
When T is held fixed, only the hidden heat term differs. If H₁ and H₂ have opposite signs, the same visible anomaly can point toward opposite immediate futures.
3. The second-order temperature equation
d2Tdt2+(d − a)dTdt+(bc − ad)T = 0
Eliminating H reveals a familiar oscillator form. The coefficient d−a controls net damping, while bc−ad supplies the restoring stiffness created by coupling feedback to memory.
4. Oscillation or direct return?
Δ = (d − a)2 − 4(bc − ad)
This is the characteristic discriminant. When Δ < 0, the eigenvalues are complex and trajectories circle in phase space. When Δ ≥ 0, the response returns or separates without oscillating. Overshoot is therefore a property of the coupled coefficients, not a decorative wiggle.
07
Physical interpretation
What the model explains physically
The model is deliberately small, but the relationships it isolates organize the exploration’s argument.
01
Growth
Surface–atmosphere feedback can reinforce an initial warm anomaly.
02
Reversal
The event changes the reservoir that supports it, creating a slower restoring influence.
03
Memory
The ocean retains information about past evolution below the visible surface.
04
Phase
T and H together distinguish whether a similar-looking event is building or declining.
05
Prediction
Subsurface observations can separate futures that surface temperature alone leaves ambiguous.
06
Irregularity
Noise and nonlinear effects change timing and amplitude without erasing every recurring relationship.
08
Guided exploration
Try these experiments
Each prompt changes one part of the mechanism. Return to the corresponding lab and make a prediction before moving the control.
01
Increase b while keeping c and d fixed. How does the phase-plane loop change?
02
Increase d. At what point does the oscillation become strongly damped or disappear?
03
Keep T(0) fixed and change H(0). How quickly do the temperature futures separate?
04
Choose a different T* in Lab B. Compare H and the instantaneous temperature tendency at the two crossings.
05
Scrub through Lab C. Which field changes sign first as the phase-space point turns?
06
Increase σ in Lab D. Does the preferred time scale remain visible?
07
Increase β and compare the largest warm anomalies with the linear model.
Exploration transition
This chapter introduced recurrence without exact periodicity. The next chapter moves from one rhythm to many interacting time scales.
Sources and further reading
J. Bjerknes, “Atmospheric Teleconnections from the Equatorial Pacific,” Monthly Weather Review, 1969.
F.-F. Jin, “An Equatorial Ocean Recharge Paradigm for ENSO. Module I: Conceptual Model,” Journal of the Atmospheric Sciences, 1997.
J. Vialard et al., “The El Niño–Southern Oscillation Recharge Oscillator Conceptual Model: Achievements and Future Prospects,” Reviews of Geophysics, 2025.
