1. Introduction
Proxy records indicate that severe climate changes in the past were induced by changes in the intensity of the Atlantic thermohaline circulation [THC; interchangeable with the Atlantic meridional overturning circulation (AMOC), especially when the overturning is primarily density driven], which was triggered by meltwater release from retreating ice sheets (Flower et al. 2011; Clark et al. 2001; Vetter et al. 2017; Clark et al. 2002). As an example, the emergence of North Atlantic Ocean warming hole in current and future climates (Kim and An 2013; Keil et al. 2020) can be analogous to a sudden temperature drop associated with AMOC shutdown during the last deglaciation (e.g., the onset of Younger Dryas). Accordingly, the impact of the North Atlantic freshwater forcing (FWF) on the AMOC has attracted considerable attention from the climate research community (e.g., Rahmstorf 1996; Clark et al. 2001; Bitz et al. 2007; Böning et al. 2016; Jackson and Wood 2018; Alkhayuon et al. 2019; Kim and An 2019; Hawkins et al. 2011; Wood et al. 2019).
A notable feature of the paleo-AMOC is that its intensity changed abruptly, indicating phase transitions between on and off states (Broecker et al. 1985; Rahmstorf 2002; McManus et al. 2004; Delworth et al. 2008; Henry et al. 2016). This nonlinearity has been explored using a hierarchy of climate models and has been understood as a hysteresis behavior with saddle-node bifurcations under increasing/decreasing FWF (Rahmstorf et al. 2005; Hawkins et al. 2011; Wood et al. 2019; An et al. 2021). In previous studies, it was often assumed that the AMOC response to FWF was close to equilibrium, because the FWF varied sufficiently slowly with time (Rahmstorf 1995; Hofmann and Rahmstorf 2009; Hu et al. 2012). However, proxy records indicate that meltwater flux in the real world may vary rapidly with time, rather than being quasi-steady with respect to the AMOC adjustment time scale (Fairbanks 1989; Aharon 2003; Wickert et al. 2013; Vetter et al. 2017). Under such time-varying forcing, a system’s response may significantly differ from that under an equilibrium, depending on how fast the system can adjust to the varying forcing. In this case, not only the adjustment time scale of the system but also the forcing time scale can be important.
Dependencies of bifurcations on the forcing time scale have been reported in other scientific fields such as acoustics, laser physics, engineering, and neuroscience (Mandel and Erneux 1987; Mannella et al. 1987; Baer and Gaekel 2008; Bergeot et al. 2014; Premraj et al. 2016; Bonciolini et al. 2018). It has been shown that time-varying forcing can postpone bifurcation of a dynamical system so that the bifurcation requires a larger parameter change beyond the static bifurcation point—that is, the parameter value at which the bifurcation would occur if the quasi-static equilibrium is satisfied (e.g., An et al. 2021). This bifurcation delay of the system is caused by its failure to adapt to the changing environment, due to inertia that keeps the system close to its initial state (Hughes et al. 2013).
In a climate system, the ocean provides such inertia. However, only a few studies to date have considered different forcing time scales in simulating the AMOC hysteresis (Alkhayuon et al. 2019; Stocker and Schmittner 1997). Stocker and Schmittner (1997, hereinafter SS1997) examined the AMOC response under CO2 forcing, which increased to a fixed level at different rates. They used a zonally averaged three-basin ocean model coupled with an energy-balance atmospheric model and found that if the CO2 is increased sufficiently slowly, the AMOC weakens at first but eventually recovers its initial strength. If the CO2 is increased rapidly, the AMOC enters a collapsed state even if the final CO2 value is the same as the slowly increased case. Alkhayuon et al. (2019, hereinafter A2019) found qualitatively similar results using a simple three-box model with freshwater flux forcing. Recently, An et al. (2021) also performed sensitivity experiments of the AMOC to the FWF time scale using H. Stommel’s two-box model and an Earth system model. Their main focus was dynamic hysteresis of the AMOC under sinusoidal FWF, whose area inside the hysteresis loop increases with the forcing frequency, indicating delayed bifurcation under rapid forcing. This study, being an extension of An et al. (2021), also explores the rate-dependent hysteresis behavior of the AMOC, but we further address the physical mechanism of how forcing rates determine the timing of the AMOC collapse. We specifically explore the relationship between accumulated forcing and bifurcation points as well as feedback processes that modulate the relationship. To this end, we conducted a sensitivity test similar to An et al. (2021) using an Earth system model.
The remainder of this paper is organized as follows: section 2 introduces the model configurations and experimental design. Section 3 describes the simulated results and elucidates the physical processes that contribute to different timings of the AMOC collapse under various forcing rates. In section 4, a comparison with previous studies is given, and section 5 summarizes the findings.
2. Methods
a. Model and experiments
We used ECBilt-CLIO-VECODE-LOCH-AGISM (LOVECLIM), version 1.3, a three-dimensional Earth system model (Goosse et al. 2010). LOVECLIM captures well the overall large-scale features in the climate with an efficient computing speed, which makes it suitable for long-term simulations that are required in this study. The atmospheric model (ECBilt), which is relatively simplified, is a spectral model that solves the quasigeostrophic potential vorticity equation (Opsteegh et al. 1998). Its horizontal resolution corresponds to 5.6°, and it has three vertical levels. The ocean model, CLIO, solves primitive equations at a horizontal resolution of 3° and vertical resolution of 20 levels (Goosse and Fichefet 1999). Freshwater volume pile-up is not explicitly simulated in the model. Instead, the freshwater forcing is treated as a negative salinity flux at the surface, so this study only analyzes the influence of surface freshening on the AMOC. The atmospheric and oceanic models are coupled to a dynamic vegetation model (VECODE; Brovkin et al. 1997) with the ice sheet (AGISM) and carbon cycle (LOCH) models deactivated.
To test the sensitivity of AMOC hysteresis on the forcing time scale, we devised FWF scenarios, as shown in Fig. 1, and applied forcing for the North Atlantic surface (50°–70°N, 15°–70°W). In type-1 scenarios, the FWF linearly increases from 0 to 1 Sv (1 Sv ≡ 106 m3 s−1) at different rates and then remains fixed. We performed 11 experiments for type 1, with rates of 0.1, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001, 0.0005, 0.0002, 0.0001, and 0.000 05 Sv yr−1, corresponding to 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10 000, and 20 000 years to 1-Sv forcing. In type 2, the FWF initially increases up to 1 Sv, similar to type 1, but decreases back at the same rate until it reaches −0.4 Sv (e.g., Hawkins et al. 2011). Seven rates of forcing were applied for the type-2 scenario; 0.001 25, 0.001, 0.000 83, 0.000 66, 0.000 56, 0.0005, and 0.000 42 Sv yr−1, corresponding to 800, 1000, 1200, 1500, 1800, 2000, and 2400 years to reach 1-Sv forcing. While type-1 experiments cover a wider range of forcing time scales, type-2 experiments cover a complete hysteresis loop. For the physical mechanism of bifurcation delay, we focus on the collapsing branch of the AMOC by using type 1. Freshwater compensation outside the forcing region was not applied so that the global mean salinity decreased as the forcing increased.
Schematic representation of forcing scenarios used in this study. In type 1, the forcing linearly increases until it reaches the predefined maximum value (1 Sv) and then remains fixed. In type 2, the forcing starts to decrease at the same rate as it was increased as soon as it hits the maximum value. The time at which the area under the curve becomes 10 m in the equivalent sea level elevation (eSLE) is denoted by t*. Note that t* and the corresponding y-axis (FWF) value vary depending on the forcing rate.
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0897.1
b. Freshwater budget
3. Results
a. Bifurcation delay
Figure 2 shows the ramp-up branches of hysteresis for type 1 and full hysteresis loops for type 2. It is noticeable that during the ramp-up period, the AMOC under fast-varying forcing collapses at a higher FWF value, indicating that a larger parameter change is required. Similarly, during the ramp-down period, the AMOC under fast-varying forcing recovers at a lower value of FWF. In other words, bifurcation is delayed so that the area inside the hysteresis loop expands as the forcing varies faster. Note that the term “delay” is in terms of the bifurcation parameter (in this case, the FWF), not the time.
Collapsing (ramp up) branch of the AMOC hysteresis for type 1 (solid) and full loops for type 2 (dashed). The AMOC index is defined as the maximum streamfunction in the North Atlantic Ocean north of 20°N and below 500 m. A 15-yr Lanczos low-pass filter was applied here for visual clarity. Bluish colors represent the static-dominant regime and reddish colors represent the dynamic-dominant regime, which is described in the text.
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0897.1
In Fig. 3, simulated bifurcation points (black dots and crosses) for the AMOC collapse are shown with relative contributions from the static and dynamic bifurcation. Here, the static bifurcation point is fixed regardless of the forcing rate, and it can be obtained by repeating a number of simulations for constant FWF values. To save computational cost, the static bifurcation point for AMOC collapse was estimated to be 0.264 Sv by an exponential fitting of dynamic bifurcations. Meanwhile, the simulated bifurcation point is measured as a point at which the 101-yr running-averaged AMOC intensity becomes lower than a predefined value. The AMOC index has a nonzero residual value by its definition even after the shutdown; thus, the criterion is considered to be 1.7 Sv. The results are not qualitatively sensitive to the filter or criterion (not shown here). The difference between the simulated bifurcation point and the static bifurcation point is termed the dynamic bifurcation, which is a measurement of the delay effect caused by the forcing. Figure 3 shows that beyond a certain value (approximately 0.0017 Sv yr−1) of the forcing rate, dynamic bifurcation becomes more dominant in determining the final bifurcation point. Although the threshold may vary depending on the criterion of the AMOC intensity, which is currently set to 1.7 Sv, a sensitivity test revealed that the threshold is insensitive to the criterion (the threshold of approximately 0.002–0.005 Sv yr−1 for the criterion of 1.7–15 Sv). For convenience, we divided the experiments into two regimes using 0.0017 Sv yr−1 as a threshold: the static-dominant regime in which forcing varies at a rate lower than 0.0017 Sv yr−1 and the dynamic-dominant regime in which forcing varies at a rate higher than 0.0017 Sv yr−1 (Fig. 3). It will be shown in the next subsections that the interactions between physical variables differ depending on dynamic or static dominance.
Relative contributions of static (blue-green shading) and dynamic (pink shading) bifurcation to the final bifurcation point (black line). The static bifurcation point was estimated on the basis of exponential fitting. Dots were obtained from the type-1 simulations, and cross marks were obtained from type 2. For this figure only, the FWF was adjusted to keep increasing up to 10 Sv for the fastest four rates (0.01, 0.02, 0.05, and 0.1 Sv yr−1) for which bifurcation is not accomplished until the FWF reaches 1 Sv. The point at which the contribution becomes the same was estimated through the interpolation and marked as a vertical line. Gray short-dashed line represents a curve of TIFWF = 10 m (see text).
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0897.1
b. Mechanisms
1) Forcing accumulation and slow adjustment of the system
In the experiments described in section 2, the control variable is freshwater flux (m3 s−1). Thus, time-integrated forcing [
Annual-mean Atlantic surface density distributions (kg m−3 − 1000) for type-1 experiments when the FWF is 0.2 Sv.
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0897.1
North Atlantic (90°W–0°, 50°–70°N) surface density in sigma (kg m−3 − 1000) vs AMOC intensity (Sv). A 100-yr Lanczos low-pass filter was applied, and the result is not sensitive to the cutoff frequency. Note that during the ramp-up period the surface density control on the AMOC intensity is almost rate independent for the static-dominant experiments (bluish) whereas it is rate dependent for the dynamic-dominant experiments (reddish).
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0897.1
Interestingly, in Fig. 5, the surface density–AMOC relationship is almost independent of the forcing rate in the static-dominant regime (bluish lines) during the ramp-up period. This implies that the AMOC response to the surface density closely follows that of the quasi-equilibrium. As the FWF lowers the surface density, the vertical density profile in the sinking region is stratified and becomes more stable (Fig. 6). Under the slow-to-moderate forcing, there is enough time for the circulation to adapt to the vertical stratification change. Thus, static bifurcation, regulated by equilibrated processes, primarily determines the bifurcation timing in this regime. Conversely, the relationship changes with the rate once it exceeds 0.002 Sv yr−1. In this regime, the forcing varies so quickly that the AMOC fails to adjust to the surface buoyancy changes. Therefore, the AMOC change and its collapse point become highly sensitive to the forcing rate (Figs. 3 and 5), causing the dynamic bifurcation to become dominant. Figure 5 also indicates that the AMOC–surface density relationship differs between ramp-up and ramp-down periods even for the same forcing rate (dashed lines) and that there arises a rate dependency in the static-dominant regime during the ramp-down. Since the cumulative effect from the ramp-up period can affect the results in the ramp-down period, we focused on the physical processes that are relevant to the AMOC collapse in this study.
Relationship between North Atlantic (50°–70°N) surface density (kg m−3 − 1000) and stratification (kg m−3) during the full hysteresis loop obtained from the type-2 experiments. The stratification is measured by the density gradient between 850 m and the surface.
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0897.1
To get further insight into the relationship between TIFWF and bifurcation point, a constant-TIFWF line (gray short-dashed) is overlaid on the simulated bifurcation points in Fig. 3, where a line for 10 m of TIFWF [in equivalent in sea level elevation (eSLE)] is shown. That is, if forcing varies at 10−4 Sv yr−1, 10 m of TIFWF is achieved when FWF is approximately 0.15 Sv. If forcing varies at 10−1 Sv yr−1, TIFWF becomes 10 m when FWF is about 4.8 Sv, and so on. The simulated bifurcation points (black solid) seem to follow this constant-TIFWF line, especially under moderate forcing rates, indicating that bifurcation tends to occur when TIFWF is about 10 m. Therefore, Fig. 3 implies that the bifurcation point is primarily determined by the total amount of freshwater input (i.e., TIFWF).
Similarly, Jackson and Wood (2018) and A2019 discussed the usefulness of time-integrated forcing as an indicator of the bifurcation point, using a state-of-the-art eddy-permitting model and a conceptual three-box model, respectively. They showed a similar result for the bifurcation point and TIFWF (e.g., Fig. 11 in A2019), but they concluded that the integrated forcing was not a useful indicator for a bifurcation point because the relationship is not exact (bifurcation points are not located “exactly” on a constant TIFWF line). However, the above analysis suggests that TIFWF plays an important role in determining bifurcation points, and the exact relationship could be modulated by other physical processes. In the remainder of this section, possible modulation processes are proposed.
2) Modulations
(i) Horizontal redistribution of the forcing
Under especially slowly increasing FWF forcing, more leakage of freshwater into other ocean basins would be expected. Such a forcing redistribution effect is illustrated in Fig. 7 as surface density distributions when the TIFWF is equivalent to the global sea level rise of 10 m (at time t* in Fig. 1 and Table 1). Note that in Fig. 7, surface density is shown as anomalies against a reference density, which is an area average in North Atlantic (Table 1), to more easily compare with the forcing area density. Under rapidly increasing forcing (upper left), the North Pacific and the Southern Ocean have much higher surface density than the North Atlantic because freshwater anomalies have not had enough time to escape the forcing area. Under slowly varying forcing (lower right), regions outside the forcing area are much fresher than fast-varying cases since the freshwater anomalies have escaped the North Atlantic and only a portion has remained. Thus, when the same amount of freshwater is given, the North Atlantic density can remain relatively high under slowly varying forcing, and more TIFWF is required to achieve the AMOC collapse. Namely, the efficiency of the given FWF to induce an AMOC bifurcation is reduced as the forcing time scale increases.
Annual-mean surface density anomalies (kg m−3) with respect to the North Atlantic (50°–70°N) area average when the TIFWF is 10 m in the eSLE. The North Atlantic area averages are given in Table 1.
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0897.1
Time to reach 10 m of TIFWF (t*) and North Atlantic (50°–70°N) surface density at t*.
Here, for a quantitative comparison of the efficiency depending on the forcing rate, we computed effective TIFWF for the type-1 experiments at time t* (Table 1). Effective TIFWF, which is estimated based on the salinity change in the Atlantic basin (see the appendix), represents the freshwater amount required for the given salinity reduction. The term “effective” is used in the following sense; if, for example, 2 m (in eSLE) of 10 m of the given freshwater amount has been exported outside the Atlantic, only the remaining 8 m can have the “effect” on the salinity reduction. The results for each experiment are shown in Fig. 8a, normalized to the most fast-varying forcing case. As expected, the effective TIFWF tends to increase with the rate, showing that under slowly increasing forcing, a large amount of the externally forced freshwater is exported to the outside of the basin. The relationship is not linear, however, with saturation occurring near the boundary between static- and dynamic-dominant regimes (about 0.002 Sv yr−1).
Scatterplots of each variable vs forcing rate when the TIFWF is 10 m. (a) Forcing efficiency measured as a relative amount of the effective freshwater in the Atlantic basin (33°S–70°N) in comparison with the 0.1 Sv yr−1 experiment; (b) oceanic circulation feedback measured as eSLE (m); (c) AMOC intensity (Sv); (d) hydrology feedback measured as eSLE (m).
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0897.1
We further compute
Figure 8b shows that the anomalous freshwater export by a combination of overturning and azonal circulations decreases with the forcing rate. The terms
In LOVECLIM, the AMOC imports freshwater into the Atlantic
Note that
Overall, the forcing redistribution plays a role in widening the hysteresis because only a portion of the given forcing can be used to reduce the Atlantic density, and more freshwater input is required for the AMOC collapse. This widening effect is more prominent in the slowly varying forcing since the additionally required freshwater input is larger (due to the low efficiency) than in the fast-varying forcing. In contrast, the hysteresis of the fast-varying forcing is least affected by its high efficiency. Therefore, different efficiencies to redistribute the forcing for different forcing rates result in a convergence of hysteresis, narrowing the gap between the experiments.
(ii) Atmospheric feedback
Here, we investigate atmospheric feedback. In all experiments, the net precipitation (precipitation minus evaporation) in the North Atlantic increases as the AMOC is weakened, which is due to a larger decrease in evaporation than in precipitation. The suppressed evaporation is attributed to surface cooling of the North Atlantic Ocean, induced by reduced meridional heat transport (not shown). Therefore, this hydrological response acts as positive feedback by adding more freshwater influx. This promotes bifurcation and narrows the hysteresis width for all experiments.
4. Discussion
The AMOC bifurcation under time-varying forcing and the associated mechanisms have been examined using an Earth system model. A delayed AMOC collapse was observed under rapidly increasing forcing, indicating that it occurred at a higher FWF value than expected in the equilibrium response. SS1997 and A2019 also investigated the forcing rate-dependent bifurcation of the AMOC. They concluded that AMOC collapse can occur even before the static bifurcation point if the forcing is increased rapidly—that is, the bifurcation can be advanced by fast increasing forcing. Different conclusions were derived between previous studies and ours because the experimental designs and objectives were different. In the studies of SS1997 and A2019, the forcing scenario took a similar form to type 1 in Fig. 1 (although the forcing was eventually turned off in A2019). However, the level of constant forcing was set to be near the static bifurcation point, and they focused on the stability of the AMOC during the constant forcing period (i.e., after adjustment). In their experiments, the AMOC response during the constant forcing period splits depending on how fast the forcing has been increased; the AMOC recovers from the weakening if the forcing has increased slowly, while it turns into a collapsed state if it has increased rapidly. The AMOC in our study, on the other hand, was shut down for all forcing rates before the FWF was fixed to its maximum value, which is far beyond the static bifurcation point. The unequilibrated AMOC response analyzed in this study, even though a highly idealized scenario was used, may be a more realistic case considering that the meltwater release from the ice sheets is likely to be more pulse-like rather than held constant (Fairbanks 1989; Aharon 2003; Wickert et al. 2013; Vetter et al. 2017). Although the interpretation of whether the fast-varying forcing provokes or delays the bifurcation depends on whether the system has adjusted or not, the simulated results are fundamentally consistent; the bifurcation is preferred when the time-integrated forcing is large. For example, in the study of A2019, tipping by rapid forcing is preferred only if the forcing is turned off sufficiently slowly or if the duration of the forcing is sufficiently long (Fig. 15 in their paper).
As a physical mechanism of the rate-dependent AMOC response, SS1997 proposed the vertical redistribution of the forcing. If the AMOC has not collapsed yet (slowly increased forcing in their study), the AMOC can recover its original strength even if the forcing has not been turned off, since buoyancy anomalies are effectively transferred to depths through convection. In contrast, once bifurcation occurs (rapidly increasing forcing), the forcing can no longer be transferred to the lower layers; thus, strong stratification and a collapsed AMOC state are maintained. This feedback was also simulated by LOVECLIM, as shown in Fig. 6. The stratification change with respect to the surface density change becomes more sensitive after the AMOC collapses as indicated by the larger slopes in the ramp-down period. During the ramp-up period, stratification is strengthened by surface freshening, but it is mitigated by vertical buoyancy transfer by the AMOC. During the ramp-down period, the AMOC was shut down and the surface density increase effectively weakened the stratification in the absence of mitigation. Taken together with the mechanism proposed in section 3, the AMOC provides feedback mainly through horizontal redistribution of forcing until its collapse and thereafter through vertical redistribution.
Although this study presents simulation results under an idealized experimental setting, it reasonably suggests the possibility of new constraints on the past ice sheet melting scenario. For example, Gregoire et al. (2016) explored the effect of the relative timing of two climatic events: the Laurentide–Cordilleran saddle collapse and Bølling warming. They showed that if the Bølling warming precedes, the saddle collapse is triggered, and the amount of accelerated melting reaches 5 m in the global mean sea level within 340 years. Conversely, if the Bølling warming does not trigger the separation of the two ice sheets (i.e., the ice sheets are too thick or saddle collapse occurs too early), melting only amounts to 2–5 m, resulting in a lower rate of release. Considering that the AMOC response to FWF depends on the forcing rate, we will be able to place more confidence in the ice sheet melting scenario that agrees with the AMOC response recorded in the proxy data.
5. Summary
In Fig. 9, the source and sink terms of the freshwater budget at TIFWF of 10 m are summarized for each experiment. Overall, the given freshwater amount in total (colored bars in the plus-sign columns) tends to increase as the forcing rate decreases because of the hydrology (sky blue) and azonal circulation (pink) effects. Despite the large input of freshwater under slowly increased forcing, the freshwater amount removed by overturning circulation (yellow) is also large, leaving a small portion of the given freshwater in the Atlantic basin (navy). The key findings associated with the physical processes for this rate dependency are summarized as follows:
The forcing accumulation in the sinking region plays an important role in determining the bifurcation point, but the exact timing of bifurcation is modulated through feedback processes by changes in the hydrology and oceanic circulations.
As the forcing increases slowly, more freshwater is required for bifurcation due to strong horizontal redistribution of the forcing, which is driven mainly by overturning circulation.
As the forcing increases slowly, anomalous precipitation, which is driven by the decreased evaporation over the cold North Atlantic surface, more effectively provides additional FWF. However, this effect is overwhelmed by finding 2.
Bar charts showing contributions from each term of Eq. (5) when the TIFWF is 10 m. Freshwater sources and sinks are marked by a plus sign and minus sign, respectively. Effective TIFWF is marked as “trapped” (navy) as it is used to lower the salinity in the basin. For comparison, all terms were converted into eSLE (m).
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0897.1
These findings can be useful from a paleoclimatic perspective because the AMOC change recorded in the proxies may provide a new constraint for ice sheet melting scenarios. In addition, under the current warming by greenhouse gas forcing, which is at an unprecedented rate, many studies have already proposed observational/theoretical evidence that the AMOC weakens as a result of anthropogenic warming (Bakker et al. 2016; Böning et al. 2016; Liu et al. 2017; Rahmstorf et al. 2005). Thus, investigating climate responses to time-varying forcing is required. However, so far, only the AMOC collapse-related mechanisms have been discussed; thus, the mechanism of the off-to-on transitions of the paleo-AMOC or reversibility of the future AMOC requires further investigation. Considering the forcing accumulation effect, type 2-like experiments may be inappropriate for discriminating the inherent physical processes of the on-to-off and off-to-on transitions. Experiments with the same initial condition of a stable AMOC-off state may be useful for this purpose and will be conducted in a future study.
Acknowledgments
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF-2017K1A3A7A03087790, NRF-2018R1A5A1024958). Author H.-J. Kim is grateful for financial support from the Hyundai Motor Chung Mong Koo Foundation. The authors also thank the editor and anonymous reviewers for providing insightful suggestions.
Data availability statement.
LOVECLIM is available online (https://www.elic.ucl.ac.be/modx/index.php?id=289).
APPENDIX
Calculation of Effective TIFWF
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