Slow and soft passage through tipping point of the Atlantic Meridional Overturning Circulation in a changing climate

The study investigates how the Atlantic Meridional Overturning Circulation (AMOC) tips under time-varying freshwater forcing, challenging the implicit assumption in classical bifurcation theory that tipping points are fixed. AMOC strongly influences global climate by transporting heat and modulating temperature, precipitation, and sea ice. Paleo evidence shows meltwater pulses can drive abrupt AMOC changes. Prior work identifies AMOC tipping behavior and thresholds across model hierarchies (from Stommel’s box model to EMICs and some fully coupled GCMs). However, many studies assume quasi-equilibrium under transient forcing and neglect the effect of the forcing rate. In non-autonomous systems, slow passage (bifurcation delay) arises when parameters change in time, potentially delaying tipping relative to the static bifurcation. The research question is how transient freshwater flux forcing (FWF) rate modulates AMOC tipping point location, timing, and abruptness, and whether such dynamics can explain observed lags in past events (e.g., MWP-1A) and inform future risk under Greenland melt.

Classical work (Stommel, 1961) established bistability and thresholds for thermohaline circulation driven by salinity/temperature contrasts. Subsequent studies found AMOC tipping points and hysteresis in conceptual models, EMICs, and some AOGCMs, including rapid collapse under elevated CO2 in a bias-corrected CCSM. Intercomparison studies highlighted model diversity and roles of ocean diffusivity and feedbacks. Dynamical systems theory recognizes multiple tipping mechanisms: bifurcation tipping (B-tipping), noise-induced tipping (N-tipping), and rate-induced tipping (R-tipping). Slow passage effects are well documented in other systems and suggest tipping delays under time-varying parameters. Prior AMOC studies have explored hysteresis, freshwater hosing, and possible R-tipping in some models, but often assume quasi-stationary adjustment. The present work builds on these by quantifying rate-dependent slow-passage delays and their implications across models and real-world forcing rates.

Two modeling frameworks were used: (1) a stochastic Stommel two-box model and (2) the EMIC LOVECLIM. In Stommel’s model, salinity difference y evolves with fixed box temperatures and Gaussian white-noise freshwater forcing; the flow strength q depends on thermal/haline expansion and a tunable exchange constant. The Fokker–Planck equation was solved numerically (central differences, implicit Euler; Δt=0.01, Δy=0.01; y∈[-0.5,2.5]) to obtain time-evolving PDFs under prescribed freshwater forcing f(t). Stationary PDFs (σ=0) yielded stable/unstable fixed points and bistable regimes (TH: AMOC on; SA: AMOC off). Stationary bifurcation points for collapse (TH→SA) and recovery (SA→TH) were f_TH≈0.74 Sv and f_SA≈0.46 Sv, with asymmetric AMOC intensity jumps (Δq_TH≈10.9 Sv; Δq_SA≈17.5 Sv). N-tipping probabilities P_TH and P_SA were computed by integrating PDFs separated by the local minimum. For transient experiments, f(t)=a+r(t−t_c) was imposed with linear rates r spanning ~1.2×10^-5 to 3.7×10^-2 Sv/year (r=10^β, β∈[-2.5,1]), for both increasing (collapse) and decreasing (recovery) cases; noise amplitudes and forcing shapes (linear vs tanh) were also tested for robustness. The transient dynamic tipping range was defined probabilistically from the evolving PDF, contrasting with the static range [f_SA,f_TH]. In LOVECLIM, linearly increasing/decreasing FWF was uniformly applied over the North Atlantic (50°–70°N, 70°–15°W) with rates r=5×10^-5 to 10 Sv/year. For r<10^-2 Sv/year, forcing increased to 1 Sv; for faster rates, up to 10 Sv to capture delayed collapse. AMOC intensity ψ was defined as the maximum overturning in the North Atlantic (>20°N), below 500 m, low-pass filtered (20-year cutoff). Tipping points were defined deterministically as the FWF level where ψ crossed 2.2 Sv (on/off threshold). Delays in tipping timing and changes in abruptness (measured as average |dψ/dt| during collapse/recovery) were computed. Analytical linkage between tipping-point delay f_delay(r) and abruptness was formulated: f_dyn(r)=f_st+f_delay(r), yielding dψ/dt≈Δψ/(f_st−f0+f_delay(r))·r and an upper bound on abruptness as r→∞ determined by df_delay/dr. LOVECLIM tipping points were fitted: collapse f_collapse=f_st+k_collapse·r (f_st=0.26 Sv, k_collapse=62.78 year; R^2=0.99); recovery f_recovery=f1+k_recovery·r+(f_st−f1)·exp(−αr) (f1=−0.2247 Sv, k_recovery=5.105 year, α=374.4 year/Sv; R^2=0.98). These fits were used to reproduce dψ/dt and estimate fast-limit abruptness. Real-world FWF rate estimates were compiled: DO events (~50–30 ka BP) from reconstructed oscillatory FWF (50–65°N, 50–10°W); MWP-1A pulse from North American ice-sheet reconstructions; Greenland runoff from GRACE (2003–2016) and downscaled CMIP6 SSP5-8.5 projections (2017–2100). Linearized rates were derived to map onto the LOVECLIM sensitivity curves.

- Slow passage (bifurcation delay) under transient freshwater forcing substantially delays AMOC tipping and softens its abruptness. - Stommel model (transient r≈0.5 ≈1.9×10^-3 Sv/year): the static tipping range [0.46,0.74] Sv shifts to a dynamic range ~[0.90,1.08] Sv; the 50–50 probability point shifts from ~0.66 Sv to ~1.04 Sv; tipping timing range shifts from ~[252,398] years to ~[488,582] years with ~170-year extra delay at the half-probability point. The tipping range narrows (width from 0.28 Sv to 0.18 Sv) and the AMOC intensity jump at boundaries decreases (Δq_lo: 17.4→10.9 Sv; Δq_up: 17.4→4.3 Sv), indicating less abrupt transitions and more continuous adjustment. Across r spanning ~1.2×10^-5–3.7×10^-2 Sv/year, dynamic tipping boundaries can increase by up to ~2.41–2.42 Sv; tipping timing delay is non-monotonic with a maximum ~1300 years near r≈3.7×10^-5 Sv/year; Δq at boundaries decreases monotonically to ~2.1 Sv at fastest r. - LOVECLIM EMIC: collapse tipping point increases approximately linearly with r from ~0.26 Sv (r=5×10^-5 Sv/year) to ~6.4 Sv (r=10^-1 Sv/year). Recovery tipping point decreases nonlinearly from ~0.21 Sv to ~−0.7 Sv over the same range. Linear sensitivity of tipping point to r is ~12× larger for collapse than recovery, indicating stronger modulation during collapse. - Tipping timing delay (LOVECLIM) is maximized around r≈2×10^-4 Sv/year at ~250 years for both collapse and recovery; delays diminish at very fast rates (collapse ~60 years, recovery ~10 years at r≥10^-1 Sv/year). - Dynamic deceleration: observed AMOC change rates are much smaller than stationary estimates. At r=10^-1 Sv/year, the stationary estimate predicts collapse at −8.17 Sv/year, while LOVECLIM simulates −0.32 Sv/year (97% reduction). For recovery, 0.12 Sv/year (static) vs 0.03 Sv/year (LOVECLIM), a 75% reduction. Analytical fits yield fast-limit abruptness bounds of ~0.32 Sv/year (collapse) and ~3.96 Sv/year (recovery). - Real-world FWF rates lie mostly in the slow regime producing large timing but modest point delays: DO events (±2.3×10^-4–1.9×10^-4 Sv/year) and MWP-1A (~2.8×10^-4 Sv/year) yield tipping point increases of ~0.05–0.07 Sv and timing delays of ~234–243 years. Greenland runoff: 2003–2016 (~1.1×10^-4 Sv/year) gives ~0.02 Sv delay in tipping point and ~224-year timing delay; 2017–2100 SSP5-8.5 (~7.2×10^-4 Sv/year) yields ~0.14 Sv tipping point delay and ~170-year timing delay. - The slow-passage delay provides a plausible mechanism for the ~1000-year lag between MWP-1A and proxy-inferred AMOC weakening; differences vs earlier hypotheses (Antarctic source) highlight alternative explanations. - Collapse tipping is more strongly delayed and decelerated than recovery, consistent with nonlinear salt-advection feedback asymmetry.

The findings show that AMOC tipping thresholds and dynamics are not fixed but depend strongly on the rate of freshwater forcing. Slow passage through bifurcations under transient forcing shifts tipping points (higher FWF needed for collapse, lower for recovery), delays tipping times, narrows tipping ranges, and reduces abruptness, especially during collapse. This dynamic modulation reconciles discrepancies between stationary hosing experiments and transient responses, and offers an explanation for observed lags in paleo records, notably the delayed AMOC response to MWP-1A. The analytical linkage between tipping-point delay and change-rate bounds clarifies why even large, rapid freshwater increases do not lead to proportionally fast AMOC changes: the delay acts as a brake, capping abruptness. The stronger modulation during collapse aligns with nonlinear salt-advection feedbacks: an active AMOC is more sensitive to freshwater perturbations than an off state is to salinity recovery. Practically, these results imply that projecting AMOC risk under accelerating Greenland melt must account for rate-dependent delays and deceleration, as they modulate both the threshold and the temporal manifestation of tipping. Inter-model differences (e.g., diffusivity) can alter the magnitude of these effects, highlighting the need for cross-model assessment.

The study demonstrates that transient freshwater forcing induces a pronounced slow-passage (bifurcation delay) effect on AMOC tipping: tipping points shift, timing is delayed (with a non-monotonic dependence on forcing rate), and abruptness is substantially reduced. This dynamic tipping modulation is robust across a stochastic box model and an EMIC and is stronger for collapse than recovery. Realistic freshwater rates from past events (DO, MWP-1A) and present/future Greenland runoff imply sizeable timing delays with modest threshold shifts, providing a potential explanation for paleo-observed AMOC lags and informing future tipping risk assessments. The analytical framework links tipping-point delay to an upper bound on AMOC change rates, offering a tool to interpret and constrain transient responses. Future work should examine inter-model variability origins (e.g., ocean mixing, feedback parameterizations), extend analyses to fully coupled GCMs and spatially resolved forcing patterns, improve reconstructions of historical FWF rates and distributions, and test for possible rate-induced tipping regimes beyond those explored here.

- Inter-model differences: LOVECLIM and conceptual models may represent feedbacks and diffusivity differently from other EMICs/AOGCMs, affecting the magnitude of tipping modulation. - Forcing idealization: Linear (and tanh) FWF scenarios and uniform spatial hosing differ from realistic, spatially heterogeneous freshwater inputs; source regions in reconstructions differ from the experiment domain. - Determination of tipping: Stommel’s model treats tipping probabilistically, while LOVECLIM uses a deterministic ψ=2.2 Sv threshold, which may influence inferred tipping points. - Parameter choices: Noise amplitude, model parameters, and thresholds (e.g., ψ criterion) introduce uncertainty in absolute values of delays and abruptness. - Real-world FWF rate estimates (DO, MWP-1A, Greenland) carry uncertainties and are simplified as linear trends; downscaled CMIP6 projections may underestimate runoff relative to observations. - Mechanistic depth: The study emphasizes phenomenology of slow passage; detailed physical mechanisms (e.g., cumulative freshwater budgets, advective redistribution, atmosphere–ocean feedbacks) are not exhaustively analyzed. - R-tipping: Rate-induced tipping was not found here; whether it occurs in AMOC under other configurations or faster/structured forcings remains unresolved.