Regular and chaotic Welander oscillations in a four-dimensional conceptual model for the Atlantic Meridional Overturning Circulation

Abstract

The Atlantic Meridional Overturning Circulation (AMOC) is a key component of the Earth's climate. Evidence indicates a twentieth-century weakening, and enhanced freshwater input to the subpolar North Atlantic may further reduce overturning strength. We present and study a conceptual four-dimensional, single-hemisphere box model with three compartments: a tropical surface box, a subpolar surface box, and a large deep-water box. Advective exchange couples the surface boxes and vertical exchange with the deep ocean is represented by a smooth convective-adjustment scheme. A comprehensive bifurcation analysis reveals an equilibrium structure with up to four coexisting overturning states, together with regimes of bistability and tristability. We identify families of periodic solutions and chaotic attractors with a clear timescale separation: a millennial oscillation is modulated by faster decadal-to-centennial variability arising from episodic shutdowns of subpolar convection. As prescribed freshwater fluxes increase, shutdown events become more frequent and the background overturning weakens. Additionally, for certain values of freshwater influx, the dynamics become chaotic, producing an irregular on-off switching of convection.

Figures (13)

• Figure 1: Hierarchy of climate models, organized by spatial dimensionality on the y-axis and by the number of climate processes on the x-axis. Adapted from dijkstra2013nonlineardijkstra2024role.
• Figure 2: Model schematic for system \ref{['eq:system_full']}, with arrows indicating flux directions and loops indicating mixing, as governed by the convective exchange functions $\mathbf{K}_E$ and $\mathbf{K}_N$. The model consists of three interconnected basins: the tropical surface box $E$ (orange), the subpolar surface box $N$ (light blue), and the deep-water box $D$ (dark blue). Each box is labeled by its volume $V_i$, temperature $T_i$, and salinity $S_i$. In the tropical box, temperature relaxes toward the atmospheric value $T_E^a$ and salinity is forced by the virtual salt flux $F_E$. In the subpolar box, temperature relaxes toward $T_N^a$ and salinity is forced by the virtual salt flux $F_N$. Meridional exchange between the tropical and subpolar boxes is driven by a density-dependent overturning function $\Psi$ and a wind-driven advection $W$.
• Figure 3: One-parameter bifurcation diagrams in the freshwater forcing $\mu$, showing branches of equilibria represented by the overturning function $\Psi$. Points $\mathrm{SN}$ of saddle--node and $\mathrm{H}$ of Hopf bifurcations are marked by black and green circles. The branches are colored to indicate their stability: stable northward (blue), stable southward (cyan), and unstable (black) equilibria. Background shading denotes $\mu$-intervals with one northward equilibrium (unshaded), one southward equilibrium (cyan), and two (purple), three (yellow), or four (red) coexisting stable equilibria. Panels (a)--(d) are for $\eta=-0.5$, $\eta=-1.5$, $\eta=-3.99$, and $\eta=-5.0$, respectively.
• Figure 4: Two-parameter bifurcation diagram of equilibria in the $(\mu,\eta)$-plane of system \ref{['eq:non_dim_system']}. Panel (a) shows a large portion of the parameter plane, with two enlargements in panels (b) and (c). Also shown are curves $\mathrm{SN}$ of saddle--node bifurcation (grey) and curves $\mathrm{H}$ of Hopf bifurcation (red; solid for supercritical and dashed for subcritical). Codimension-two points along these curves are indicated: Bogdanov--Takens $\mathrm{BT}$ (red), generalized Hopf $\mathrm{GH}$ (brown), and zero--Hopf $\mathrm{ZH}$ (cyan). The curves $\mathrm{SN}$ and $\mathrm{H}$ delineate regions with different numbers and types of stable equilibria: one northward equilibrium (unshaded); one southward equilibrium (cyan); two equilibria (purple); three equilibria (yellow); and four equilibria (red). The grey lines correspond to $\eta$-values used in Figure \ref{['fig:EqCurves']}.
• Figure 5: Geometric classification of mixing regimes along a periodic orbit $\Gamma$ representing a Welander oscillation in system \ref{['eq:non_dim_system']} for $(\mu,\eta)=(-2.11\times10^{-3},-3.99)$. Panel (a) shows $\Gamma$ in projection onto the surface of the convective exchange function $\mathbf{K}_N$ over the $(\rho_B,\rho_N)$-plane, which is shaded by mixing regime: convective (red), diffusive (blue), and a narrow intermediate regime (semi-transparent). The boundaries between these regimes are the curvature loci $L^{+}$ and $L^{-}$ of maximal and minimal curvature. Panel (b) shows the corresponding time series $\mathbf{K}_N(t)$ along $\Gamma$; orange dots mark the minimum value of $\mathbf{K}_N$ during each convective shutdown event.