Fig. 3: The tipping modulation experiment with varying freshwater forcing rate using Stommel’s model.

a A tipping range is plotted with its boundaries, \({{{\mathrm{f}}}}_{{{{\mathrm{lo}}}}}^ \ast\) (lower black line) and \({{{\mathrm{f}}}}_{{{{\mathrm{up}}}}}^ \ast\) (upper black line). The red color denotes \({{{\mathrm{p}}}}_{{{{\mathrm{SA}}}}}\), the probability for being in the tipped state. b A tipping timing delay for \({{{\mathrm{f}}}}_{{{{\mathrm{lo}}}}}^ \ast\) (blue line) and \({{{\mathrm{f}}}}_{{{{\mathrm{up}}}}}^ \ast\) (orange line). The time unit is years. c The left axis denotes difference of y at the tipping range boundaries, \(\Delta {{{\mathrm{y}}}}_{{{{\mathrm{lo}}}}}^ \ast\) (blue thick line) and \(\Delta {{{\mathrm{y}}}}_{{{{\mathrm{up}}}}}^ \ast\) (orange thick line). The right axis denotes y values at the tipping range boundaries, \({{{\mathrm{y}}}}_{{{{\mathrm{lo}}}}}^ \ast\) (blue dashed line) and \({{{\mathrm{y}}}}_{{{{\mathrm{up}}}}}^ \ast\) (orange dashed line). On the collapse case, a, b and c, the thin gray line is shown at r = 0.5 (approximately \(1.9 \times 10^{ - 3}{{{\mathrm{Sv}}}}/{{{\mathrm{year}}}}\)). The cases shown in Fig. 2d, e, and f are the same as those shown in a, b, and c, respectively, but for the recovery case.