Tipping Bifurcations in Conceptual Ocean Circulation Models

TL;DR

This study reframes AMOC tipping by treating the meridional temperature gradient as a dynamic control parameter within a two-box Stommel-type framework. By reformulating Cessi's model to couple thermal and haline forcing and projecting onto the fast temperature manifold, the authors derive a one-dimensional salinity dynamics and apply cusp catastrophe analysis to map a two-parameter stability surface. They establish the existence of a cusp bifurcation that organizes the transition between bistable and monostable regimes, with the geometry arising from the interplay of temperature-driven and salinity-driven feedbacks. The work highlights multi-parameter tipping pathways in thermohaline circulation, showing that polar amplification and thermal erosion can reshape stability and tipping susceptibility beyond traditional freshwater-focusing scenarios.

Abstract

The Atlantic Meridional Overturning Circulation (AMOC) is often analyzed using low-order box models to understand tipping points. Historically, these studies focus on freshwater flux as the primary bifurcation parameter, treating the temperature gradient as a fixed restoring target. However, the erosion of the equator-to-pole temperature contrast due to polar amplification suggests that thermal forcing should be treated as a dynamic control parameter. In this study, we use Cessi's reduced box model to map the global bifurcation structure of the thermohaline circulation. We relax the assumption of a fixed thermal background and analyze the system's behavior under joint thermal and haline forcing. We prove the existence of a cusp bifurcation, identifying the specific geometry of pitchfork and saddle-node bifurcations that bound the stable regime. This geometric characterization reveals that thermal erosion acts as a distinct mechanism for destabilization, capable of driving the system across critical thresholds even in the absence of anomalous freshwater forcing.

Figures (7)

• Figure 1: Potential energy landscapes for bistable ($P=4.98$) and monostable ($P=5.89$) freshwater forcing regimes, illustrating the potential barrier separating the competing stable states.
• Figure 1: 3D potential energy landscape $V(y, \theta)$, visualizing how the depth and number of stability wells evolve as the equilibrium temperature gradient $\theta$ varies.
• Figure 1: The discriminant surface $\Delta(\theta, P)$ in parameter space. The zero-level contour (black curve) defines the fold curves that bound the region of bistability.
• Figure 1: The discriminant surface generated by parameters $P$ and $\theta$.
• Figure 2: Bifurcation diagram showing the stable (solid) and unstable (dashed) equilibrium branches of the circulation as a function of freshwater forcing $P$. The upper and lower stable branches correspond to salinity-dominated and thermally-dominated modes, respectively.

Theorems & Definitions (18)

• Lemma 3.1: Model Reformulation
• Theorem 3.2: Temperature Induced Hysteresis
• Lemma 4.1: Tschirnhaus Transformation
• Corollary 4.2: Depressed Dynamics
• Lemma 4.3: Discriminant of Salinity Dynamics
• Theorem 4.4: Cusp Theorem
• Lemma 5.1
• Proof 1
• Theorem 6.1
• Proof 2