Fig. 5: Critical transition in barrier’s elevation and the onset of potential barrier loss driven by SLR.

a Simulations of the steady-state stochastic dynamics, represented by the PDF \({f}_{h}(h| {\lambda }_{0}^{+},{\overline{S}}^{+})\) of barrier elevation h sampled four times per year (symbols), over the phase curve shown by the dashed line in Fig. 4. The parametric phase curve (\({\lambda }_{0}^{+}({\overline{h}}_{0}),{\overline{S}}^{+}({\overline{h}}_{0})\)) is a function of mean base elevation \({\overline{h}}_{0}(t)\), which itself changes with time due to SLR as \({\overline{h}}_{0}(t)=1.3\,{{{\rm{m}}}}-R\,t\). The rate R of relative SLR is assumed to be constant and equal to the average for the intermediate scenario estimated for the region from 2020 to 2050 (R = 10 mm/yr)⁴³. All other parameters are taken as for Hog Island (Table 1). The rescaled dune recovery time \({\overline{T}}_{{\rm {r}}}^{+}\) over the phase curve (solid line) shows the critical slowing down of the dynamics as it approaches the critical transition to the low-barrier state, represented qualitatively by a tipping point. b–e Approximate basin of attraction of the two most probable equilibrium elevations, the mean maximum dune height \(\overline{H}\) and the mean base elevation \({\overline{h}}_{0}\), represented by the inverted PDF f_h(h) and interpreted as a potential function⁴⁴.