The Atlantic Meridional Overturning Circulation (AMOC) is a crucial component of the Earth's climate system, transporting vast amounts of heat northward and influencing global temperature, rainfall, and sea ice. Paleo-proxy data reveals that past meltwater pulses from ice melt have triggered AMOC variations and abrupt climate shifts. Future climate simulations also indicate that a weakened AMOC can significantly alter surface warming patterns. A key characteristic of the AMOC is its susceptibility to abrupt change at a certain threshold, a 'tipping point'. Stommel's box model provides a classic illustration, predicting an abrupt transition between stable states based on freshwater forcing at a bifurcation point. This tipping point behavior is observed across various AMOC models, including conceptual models, Earth system models of intermediate complexity, and even some fully coupled global climate models. However, many studies assume a quasi-equilibrium state under transient forcing, neglecting the rate of external forcing. This approximation is inadequate for a dynamically changing climate. When considering time-dependent forcing, the system becomes non-autonomous, introducing phenomena like bifurcation delay or the slow passage effect. This effect delays the bifurcation point when the system cannot instantaneously adjust to a time-varying parameter. This study investigates AMOC tipping under transient freshwater forcing across a wide range of timescales, using Stommel's model and an Earth system model of intermediate complexity to analyze the impact of the freshwater forcing rate on AMOC tipping and its modulation in past, present, and future climates.

Numerous studies have explored the tipping point behavior of the AMOC, using various modeling approaches. Early work, such as Stommel's box model, established the theoretical framework for understanding abrupt transitions in the circulation. More complex models, including Earth system models of intermediate complexity and fully coupled global climate models, have corroborated the existence of tipping points under various forcing scenarios, such as increased atmospheric CO2. However, a common limitation in past research has been the assumption of quasi-equilibrium states under transient forcing. This simplification often overlooks the critical role of the rate of change of external forcing, particularly in the context of a dynamically changing climate. The concept of bifurcation delay or the slow passage effect, well-established in other fields, has been relatively less explored in the context of AMOC dynamics. This paper directly addresses this gap by systematically investigating the impact of time-varying freshwater forcing on the AMOC tipping behavior.

This study employed two distinct modeling approaches to investigate AMOC tipping under transient freshwater forcing: Stommel's box model and the LOVECLIM Earth system model of intermediate complexity. **Stommel's Model:** This simplified two-box model captures the essential dynamics of the AMOC, driven by salinity and temperature differences between low- and high-latitude ocean boxes. Stochastic forcing was added to represent short-timescale variability. The model's non-dimensional salinity difference equation was solved using the Fokker-Planck equation, with the freshwater flux (FWF) set as a linearly increasing function with varying rates (r). The time evolution of the probability density function (PDF) for the flow intensity was calculated. Stationary PDF solutions were also computed for comparison. Dimensionless quantities were converted to dimensional values using scaling factors derived from literature. The probabilities of being in the 'AMOC on' (TH) and 'AMOC off' (SA) states were calculated to quantify noise-induced tipping. The tipping range (between the lower and upper boundaries of FWF) and timing were determined, and the abruptness of tipping was quantified by the difference in flow intensity (Δq) between the stable states. **LOVECLIM Model:** The LOVECLIM model is a three-dimensional Earth system model. A series of experiments were performed with varying FWF rates applied to a specific North Atlantic region (50-70°N, 70-15°W), simulating both AMOC collapse and recovery. The FWF was set as a linearly increasing/decreasing function. Tipping points were identified as the FWF levels where the AMOC intensity (maximum overturning stream function) crossed predefined thresholds. A low-pass filter was applied to the AMOC intensity time series to remove high-frequency variability. The average AMOC intensity change rate during collapse and recovery was calculated to assess abruptness. The dynamic deceleration effect (reduction in AMOC change rate due to transient FWF) was quantified by comparing the simulated AMOC rate with a static response. To link the dynamic deceleration effect with the tipping delay, a mathematical formulation was developed. The delayed dynamic tipping point (f_dyn) was expressed as a function of the static tipping point (f_st) and a delay term (f_delay) dependent on the FWF rate (r). The abruptness of tipping (dψ/dt) was formulated, demonstrating that the tipping delay reduces the abruptness. The maximum abruptness at fast FWF limits was derived using L'Hôpital's rule. LOVECLIM results were used to estimate the parameters of the formulation and reproduce the abruptness. FWF rates for past (Dansgaard-Oeschger events, Meltwater Pulse 1A), present, and future (Greenland ice sheet melting) climates were estimated using various data sources (reconstructions, observations, and climate model projections). These rates were used to illustrate the dynamic tipping modulation in different climate contexts.

This study revealed several key findings regarding the dynamic tipping modulation of the AMOC under transient freshwater forcing: 1. **Significant Tipping Point Delay:** Both Stommel's model and LOVECLIM demonstrated a significant delay in the tipping point as the rate of FWF increased. In Stommel's model, the tipping range shifted significantly, requiring higher FWF levels for collapse. LOVECLIM showed a linear increase in the collapse tipping point with increasing FWF rate and a non-linear decrease in the recovery tipping point. The linear sensitivity of collapse tipping point to r was about 12 times larger than the recovery point's. 2. **Weakened Abruptness:** The abruptness of AMOC tipping was significantly weakened under transient forcing. In Stommel's model, the difference in flow intensity (Δq) between stable states decreased with increasing FWF rates, indicating a less abrupt transition. Similarly, the LOVECLIM results showed a significant decrease in the average AMOC intensity change rate during both collapse and recovery with increased FWF rates, with a more pronounced effect for collapse. 3. **Delayed Tipping Timing:** The timing of AMOC tipping was also significantly delayed. In Stommel's model, this delay was more pronounced in the slow FWF regime and showed a non-monotonic relationship with the FWF rate. LOVECLIM showed that both the collapse and recovery times were delayed in the slow FWF regime and less delayed in the fast FWF regime, and they were maximized at the same point. This delay resulted from the competing effects of an increased distance to the tipping point and the increased rate at which that point is approached. 4. **Asymmetric Modulation:** The dynamic tipping modulation was much stronger for AMOC collapse than for recovery. This asymmetry was evident in both the magnitude of tipping point delay and the reduction in abruptness. 5. **Robustness:** Sensitivity experiments in Stommel's model showed that the tipping delay was robust across different FWF noise amplitudes and forcing scenarios. 6. **Relevance to Past, Present, and Future Climates:** Analysis of FWF rates from Dansgaard-Oeschger events, Meltwater Pulse 1A, and Greenland ice sheet melting demonstrated the applicability of the dynamic tipping modulation framework. The estimated FWF rates for DO and MWP-1A events fell within the slow FWF regime, characterized by large tipping timing delays and small tipping point delays. The large tipping timing delay may potentially explain the observed lag between MWP-1A and the subsequent AMOC change. The projected increased Greenland meltwater runoff could lead to a weakened tipping timing delay effect.

The findings of this study provide crucial insights into the dynamics of AMOC tipping under time-varying freshwater forcing. The identified dynamic tipping modulation—the significant delay in the tipping point, reduced abruptness, and delayed timing—challenges the traditional understanding of AMOC thresholds based on static bifurcation theory. The results suggest that the rate of change in freshwater forcing is a critical factor influencing the likelihood and timing of AMOC tipping. This has important implications for assessing the risk of future AMOC collapse, as accelerated Greenland ice sheet melt increases the FWF rate. The model results highlight a complex interplay between the rate of forcing and the system's response time, which may lead to counterintuitive behavior such as a larger tipping time delay even when the tipping point is not significantly delayed. The asymmetry in the dynamic tipping modulation between collapse and recovery is likely linked to the non-linear feedback processes within the AMOC system. The observed delayed tipping in the AMOC may also be responsible for the observed lagged response of the AMOC to events like MWP-1A. The study also shows a potential for inter-model differences in the tipping modulation effect due to the differences in ocean model diffusivities. Finally, the study's findings are relevant to other scenarios involving overshooting tipping points, suggesting that the delay mechanism could lead to a situation where the system does not necessarily tip even if initially passing the static tipping point threshold.

This research demonstrates the significant influence of transient freshwater forcing rates on AMOC tipping behavior. The dynamic tipping modulation effect, characterized by delayed tipping points, reduced abruptness, and complex timing delays, should be considered in future AMOC projections and assessments of climate tipping risks. The framework presented provides a more nuanced understanding of AMOC dynamics in a changing climate. Further research should focus on improving the accuracy of FWF rate estimations, investigating the role of other factors (e.g., atmospheric feedback) on dynamic tipping modulation, and exploring the potential for rate-induced tipping in AMOC models. Improved representation of AMOC dynamics in climate models is essential for accurate predictions of future climate change.

The study's conclusions are based on simplified models (Stommel's model and LOVECLIM). While these models capture essential features of the AMOC, they may not fully represent all the complex interactions within the real climate system. The estimation of FWF rates for past events relies on reconstructions with inherent uncertainties. Furthermore, the LOVECLIM model imposed FWF in a spatially uniform manner over a specific North Atlantic region, while actual freshwater input is more complex spatially and temporally. This simplified forcing could lead to some degree of underestimation or overestimation of the modulation effects. It is possible that there are other forcing factors that could affect the tipping point. Finally, the study did not investigate potential rate-induced tipping, which could occur under extremely rapid forcing scenarios.