Global stability of the Atlantic overturning circulation: Edge state, long transients and boundary crisis under CO$_2$ forcing

Using an intermediate-complexity climate model, this study reveals that the Atlantic Meridional Overturning Circulation (AMOC) undergoes a boundary crisis under rising CO$_2$ levels, where the collapse of its stable state and the resulting long chaotic transients governed by edge states explain large ensemble variances and apparent stochastic bifurcations in Earth system models.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Picture: The Ocean's "Heartbeat"

Imagine the Earth's climate system as a giant, complex machine. One of its most important parts is the Atlantic Meridional Overturning Circulation (AMOC). You can think of the AMOC as the giant conveyor belt or the heart of the Atlantic Ocean. It pumps warm water from the tropics up to Europe (keeping it mild) and sends cold, salty water back down to the south.

Scientists have long worried that if we pump too much carbon dioxide (CO2) into the atmosphere, this "heart" might stop beating or switch to a very weak, slow rhythm. This would be a disaster for the climate.

The problem is: We don't know exactly when or how this will happen.

The Problem: Why is it so hard to predict?

Usually, when scientists try to predict a disaster, they look at how a system behaves when it's close to a breaking point. They look for "warning signs," like a car engine sputtering before it dies.

However, this paper argues that looking at the "engine" (the stable state) isn't enough. The Earth is a chaotic system. Even if two groups of scientists run the exact same simulation with the exact same future CO2 levels, one group might predict the ocean current will stay strong, while the other predicts it will collapse.

Why? Because the system is sitting on a knife-edge.

The New Idea: Finding the "Edge State" (The Melancholia State)

The authors of this paper decided to stop looking only at the "Strong" state (the healthy ocean) and the "Weak" state (the collapsed ocean). Instead, they looked for the Edge State.

The Analogy: The Mountain Pass
Imagine a landscape with two deep valleys:

1. Valley A (The Strong Ocean): A deep, comfortable valley where the ocean is healthy.
2. Valley B (The Weak Ocean): A different deep valley where the ocean has collapsed.

Between these two valleys is a high, narrow mountain pass.

• If you stand in Valley A, you are safe.
• If you stand in Valley B, you are stuck in the weak state.
• The Edge State is the mountain pass itself.

The mountain pass is unstable. If you stand there, a tiny breeze (a small change in weather) will push you down into one valley or the other. You can't stay there forever. But, if you are on the pass, you are right on the boundary between life and death for the ocean current.

The researchers used a special computer trick (called "edge tracking") to find this invisible mountain pass in their climate model. They found that while the pass is unstable, the ocean can get stuck "wandering" along this pass for hundreds of years, oscillating back and forth before finally falling into the weak valley.

The Discovery: The "Ghost" and the Crisis

The researchers tested what happens as they increased the CO2 in their model, simulating global warming.

1. At Low CO2: The mountain pass exists. The ocean can wander near it, but it usually stays in the "Strong" valley.
2. At High CO2: The mountain pass disappears! The "Strong" valley and the "Mountain Pass" crash into each other and vanish.

This is called a Boundary Crisis.

The Analogy: The Ghost
When the Strong Valley and the Pass merge and disappear, they leave behind a Ghost.

• Imagine you are driving on a road that suddenly vanishes. For a while, your car keeps driving as if the road is still there (because of momentum), but eventually, you will fall off the cliff.
• In the model, once the CO2 gets too high, the ocean current enters a "Ghost State." It looks like it's still strong, and it might even oscillate wildly for a few hundred or thousand years. But it is actually doomed. It is just a "ghost" of the strong current, waiting to collapse.

Why Does This Matter? (The "Split" in the Ensemble)

This explains a confusing thing scientists see in big climate models. When they run 10 simulations with the same future CO2 scenario:

• Some simulations show the ocean staying strong.
• Some show it collapsing immediately.
• Some show it collapsing after 500 years.

The Explanation:
Because the system is near this "Ghost" or "Edge," the tiny, random internal fluctuations of the weather (like a slightly stronger wind or a slightly warmer summer) act like that "breeze" on the mountain pass.

• One simulation gets pushed down into the "Weak" valley immediately.
• Another gets pushed back into the "Strong" valley.
• A third gets stuck wandering on the "Ghost" path for centuries before finally falling.

This isn't a mistake in the model; it's a fundamental property of a chaotic system near a tipping point.

The Conclusion: What Should We Do?

The paper concludes that we cannot rely on simple "early warning signs" (like the engine sputtering) because the system might not sputter; it might just suddenly vanish into a "Ghost" state.

1. The Tipping Point is a "Window," not a Line: It's not a single moment where the ocean stops. It's a range of CO2 levels where the system becomes incredibly unpredictable.
2. The "Ghost" is Dangerous: Even if the ocean looks strong, if we cross the crisis point, it might be a "Ghost" that will collapse centuries later, giving us a false sense of security.
3. We Need a New View: To understand the future, we need to map the "mountain passes" (the edge states) and the "ghosts," not just the valleys.

In short: The Atlantic Ocean is like a tightrope walker. We used to think we just needed to watch the walker's feet to see if they were slipping. This paper says we need to look at the wind, the rope's tension, and the invisible "ghosts" of the rope that might snap without warning. If we push the CO2 too high, the rope disappears, and the walker falls, even if they looked fine for a long time.

Technical Summary

Here is a detailed technical summary of the paper "Global stability of the Atlantic overturning circulation: Edge state, long transients and boundary crisis under CO2 forcing."

1. Problem Statement

The Atlantic Meridional Overturning Circulation (AMOC) is a critical component of the Earth's climate system, responsible for transporting heat northward. There is growing concern that anthropogenic warming could push the AMOC past a tipping point, causing a transition to a weak or collapsed state (OFF state) with severe global climate consequences.

Current methods for assessing this risk rely heavily on local stability analysis and early warning signals (EWS) based on "critical slowing-down" near equilibrium. However, these methods have significant limitations:

• They assume the system remains close to a stable equilibrium, which may not hold under rapid, time-dependent forcing.
• They fail to explain ensemble splitting, where identical time-dependent forcing scenarios produce divergent outcomes (some trajectories tip, others do not) due to internal variability.
• They overlook the role of unstable states (non-attracting invariant sets) that govern transient dynamics and critical transitions in chaotic, multistable systems.

The paper aims to address these gaps by adopting a global dynamical systems perspective to map the stability landscape of the AMOC, specifically investigating the role of edge states and boundary crises under varying CO2 concentrations.

2. Methodology

The authors utilized PlaSim-LSG, an Earth system model of intermediate complexity (coupled atmosphere, ocean, sea ice, and hydrology) with approximately $10^5$ degrees of freedom.

• Experimental Setup:
  • Simulations were run under fixed CO2 concentrations (285 ppm, 360 ppm, 460 ppm) to analyze "frozen" systems.
  • Time-dependent forcing experiments were conducted using Shared Socioeconomic Pathways (SSP1-2.6, SSP2-4.5, SSP4-6.0) extending to the year 3000.
• Edge Tracking Algorithm:
  • To locate the edge state (a chaotic saddle on the basin boundary separating the strong ON and weak OFF attractors), the authors employed an iterative edge tracking algorithm.
  • This involves bisecting the state space between initial conditions converging to the ON and OFF states. By repeatedly refining the initial conditions and running parallel simulations, the algorithm converges to a "pseudotrajectory" that shadows the unstable basin boundary.
• Reduced State Space Analysis:
  • To visualize the high-dimensional dynamics, the authors constructed a reduced state space using three physically meaningful variables derived from zonally averaged Atlantic salinity:
    1. Meridional salinity gradient (SG).
    2. Vertical SG in the North Atlantic.
    3. Deep North Atlantic salinity anomaly.
• Comparative Analysis:
  • Results from PlaSim-LSG were compared with simulations from the more complex GISS-E2-1-G model (CMIP6) under the same SSP2-4.5 scenario to validate the universality of the findings.

3. Key Contributions

• Explicit Computation of the Edge State: The study successfully computed the edge state (or "Melancholia state") for a fully coupled climate model. This state acts as a "mountain pass" between the stable ON and OFF attractors.
• Discovery of a Boundary Crisis: The authors identified a boundary crisis occurring at CO2 levels (around 460 ppm) where the stable ON attractor collides with the edge state and disappears.
• Characterization of the "Ghost State": Following the boundary crisis, the system enters a monostable regime (only the OFF state is stable), but the former ON and edge states merge into a ghost state. This is a long-lived, unstable chaotic set that can trap trajectories for centuries or millennia before they eventually collapse.
• Explanation of Ensemble Splitting: The paper provides a dynamical systems explanation for the "stochastic bifurcation" (ensemble splitting) observed in complex models. It demonstrates that splitting occurs because trajectories interact with the edge state/ghost state region, where internal variability determines whether a trajectory escapes to the OFF state or remains in a transient oscillatory phase.

4. Key Results

A. Dynamics of the Edge State (at 360 ppm)

• Centennial Oscillations: The edge state is not a static point but a chaotic saddle exhibiting large-amplitude oscillations with a period of approximately 118 years and an amplitude of up to 10 Sv.
• Physical Drivers: These oscillations are driven by a complex interplay of atmosphere-ice-ocean interactions in the North Atlantic, specifically involving:
  • Sea ice retreat in the Labrador Sea (LabS) allowing deep convection.
  • Salt-advection feedback amplifying the circulation.
  • Freshwater influx from the Arctic via the Denmark Strait disrupting convection.
• Excursive Observables: The edge state exhibits "excursive" behavior, meaning certain variables (e.g., deep North Atlantic salinity, specific sea ice patterns) lie outside the range bounded by the ON and OFF states. This suggests that standard indicators might miss the approach to a tipping point.

B. The Boundary Crisis and Ghost State (at 460 ppm)

• Collision: As CO2 increases to 460 ppm, the ON attractor collides with the edge state. The ON state loses stability, and the system becomes monostable (only the OFF state is asymptotically stable).
• Long Transients: Trajectories initialized near the former ON state enter a ghost state. They exhibit long, chaotic transients (lasting thousands of years) characterized by alternating oscillations reminiscent of the ON and edge states before finally collapsing to the OFF state.
• Unpredictability: In this regime, the predictability of the asymptotic state is severely limited. Identical forcing can lead to vastly different transient durations (from a few hundred to several thousand years) depending on internal variability.

C. Response to Time-Dependent Forcing (SSPs)

• SSP1-2.6 (Low Emissions): The system remains in the bistable regime; the ON state persists.
• SSP4-6.0 (High Emissions): The forcing rate is so rapid that the system passes through the edge state region too quickly to exhibit complex transients, collapsing directly to the OFF state.
• SSP2-4.5 (Intermediate Emissions): This scenario triggers ensemble splitting. Some ensemble members collapse, while others persist or oscillate for centuries. The reduced state space analysis shows that splitting members traverse the region of the edge state/ghost state, confirming the mechanism of rate-dependent tipping.

D. Comparison with GISS Model

• The authors mapped GISS model trajectories onto the PlaSim-LSG reduced state space.
• The GISS ensemble members exhibiting divergent behavior (some recovering, some collapsing) followed paths qualitatively similar to the edge state and ghost state trajectories found in PlaSim-LSG.
• This suggests that the "stochastic bifurcation" observed in high-complexity CMIP6 models is a signature of the same underlying global stability structure (edge states near a boundary crisis).

5. Significance

• Theoretical Advancement: The study moves beyond local stability analysis to a global view of climate stability, demonstrating that unstable invariant sets (edge states) are crucial for understanding transient climate behavior.
• Explanation of Model Discrepancies: It resolves the mystery of why different climate models (and even different members of the same model ensemble) show divergent AMOC responses under identical forcing, attributing it to the chaotic nature of the basin boundary.
• Implications for Risk Assessment: The existence of long transients and ghost states implies that the AMOC could appear stable for centuries even after crossing a critical threshold, only to collapse suddenly later. This challenges the reliability of current tipping time estimates based on critical slowing-down.
• Early Warning Systems: The identification of "excursive observables" (variables that behave non-monotonically or outside the range of stable states) offers new potential targets for detecting critical transitions before they occur.
• Policy Relevance: The boundary crisis in the model occurs at CO2 levels (approx. 460 ppm) that could be reached within decades under intermediate emission scenarios, highlighting the urgency of understanding non-linear, rate-dependent tipping mechanisms.

In conclusion, this paper establishes that the stability of the AMOC is governed by a complex landscape involving unstable edge states. Under increasing CO2, the collision of these states leads to a boundary crisis, creating a "ghost state" that induces long, unpredictable transients and ensemble splitting, fundamentally altering how we must approach climate risk assessment and prediction.