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Recharge-oscillator parameters
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The science
The model
Following the recharge-oscillator idea (Jin 1997), the eastern-Pacific sea-surface temperature anomaly T (a Nino-3.4-like index, C) and the basin-mean thermocline depth anomaly h (a proxy for equatorial heat content, m) obey dT/dt = R T + F h - e T^3 (+ noise) and dh/dt = - a T - r h. Positive R is the Bjerknes feedback: warm east, weaker trade winds, less upwelling of cold water, warmer east. F h turns a deeper thermocline into warming; -a T says El Nino discharges the warm-water volume off the equator; -r h is damping.
Period and growth from the eigenvalues
With e = 0 and no noise the matrix has trace R - r and determinant F a - R r. The eigenvalues are sigma +/- i omega with sigma = (R - r)/2 and omega = sqrt(F a - ((R + r)/2)^2), so the period is 2 pi / omega. Multi-year cycles arise because F a is a product of two slow rates. The oscillation grows if R > r. The page checks this against a simulated run and against the exact matrix-exponential solution.
Nonlinearity and noise
The cubic term caps the amplitude near sqrt(R/e) for small damping. Random westerly wind bursts enter as white noise scaled by sqrt(dt) (Euler-Maruyama), which turns a damped system into irregular, sustained events.
What the 3D scene shows
The thermocline sheet follows depth(x) = 100 m + h + 50 m (1 - 0.3 T)(1 - 2x), deep in the west, shallow in the east, flatter when T is large. SST colour, trade-wind arrows (weaker for T > 0), the convection clouds (shifting east in El Nino) and the upwelling streaks are drawn from the same two numbers T and h. The vertical scale is strongly exaggerated and only the top 250 m are shown.
Honest limits
This is a two-variable caricature, not a climate model: no seasonal cycle, no spatial structure, no Kelvin or Rossby wave propagation, no El Nino / La Nina asymmetry beyond the cubic term, and the constants are illustrative tuned values, not fits to data. It cannot forecast real events.