A unified account of Earth's climate variability: dominant modes as eigenmodes of one stochastic operator

Abstract. The Madden–Julian Oscillation, the El Niño–Southern Oscillation, the Indian Ocean Dipole, the Atlantic zonal mode and the annular modes are conventionally explained by separate theories with separate parameters. We show that they can emerge from one minimal description. The climate state is modelled as four fields on the sphere, an atmospheric streamfunction, a column moisture anomaly, an ocean mixed-layer temperature and an upper-ocean heat content, whose joint evolution is written as a single linear stochastic equation. The operator contains rotation, dissipation, air–sea and moisture coupling, and the geography of the continents, and no mode-specific physics. Its eigenvalue spectrum separates into three families without any tuning of individual modes: damped atmospheric patterns with e-folding times of about a week, an eastward-propagating equatorial moisture mode with a period of 47 days, and basin-confined coupled oscillators with periods of years. Integrating the system under stochastic forcing reproduces the observed spectral bands, with peaks at 4.29 yr for Niño 3.4, 1.54 yr for the Dipole Mode Index, 1.54 yr for the Atlantic index and 69.1 days for the moisture index, together with red backgrounds for the annular indices. The same periods obtained from the resolvent of the autonomous operator, which requires no integration and carries no sampling error, are 4.06 yr, 1.54 yr and 1.54 yr. The interannual periods are not fitted: they follow from the zonal width of each ocean basin, with a fitted exponent of 1.00 on the resolvent peaks and 1.07 on the eigenmode periods, and they reproduce the observed ordering and approximate spacing of the Pacific, Indian and Atlantic modes. Four natural forcing terms are then added, followed by an anthropogenic one. Annual modulation of the coupling produces seasonal phase locking of approximately the observed magnitude. Stochastic forcing produces the red spectra of the extratropical modes without any oscillatory mechanism. An imposed volcanic cooling produces a damped oscillatory Pacific response with a rebound near two years and a memoryless extratropical response. Slow modulation of the dissipation gives the mode variances a common low frequency component. Anthropogenic forcing is treated last and separates into two channels: an additive radiative term, which by superposition cannot alter any anomaly statistic, and a parametric dependence of the operator on the background state, which can. Parameterising the second through five physical channels gives a shortening of the interannual periods governed by the increase in equatorial wave speed alone, so that the fractional period change equals minus the fractional change in wave speed, and an amplitude response that is the residue of four opposing contributions of comparable magnitude. The second result predicts that projections of ENSO amplitude should be small and of uncertain sign, which is what the data show.