Modelling dynamic strains on ice shelves resulting from flexural and extensional motions forced by ocean wave packets

TL;DR

The paper develops a thin-plate, two-dimensional model of an Antarctic ice shelf that supports both flexural and extensional waves and is forced by incident ocean wave packets. By combining a depth-averaged water-ice coupling in the frequency domain with a time-domain reconstruction, it shows that extensional waves can contribute significantly to shelf strains, particularly when flexural and extensional waves interact coherently during transient forcing. The study finds that maximum strains arise from phase-coherent interactions, matching or exceeding predictions from frequency-domain analyses, and highlights the importance of accounting for extensional dynamics in swell-driven shelf response. These results underscore the need to incorporate extensional waves and spatial variability (thickness, crevasses) in predictive models for Antarctic ice-shelf calving and stability under increasing swell exposure.

Abstract

The transient response of an ice shelf to an incident wave packet from the open ocean is studied with a model that allows for extensional waves in the ice shelf, in addition to the standard flexural waves. Results are given for strains imposed on the ice shelf by the incident packet, over a range of peak periods in the swell regime and a range of packet widths. In spite of the large difference in speeds of the extensional and flexural waves, it is shown that there is generally an interval of time during which they interact, and the coherent phases of the interactions generate the greatest ice shelf strain magnitudes. The findings indicate that incorporating extensional waves into models is potentially important for predicting the response of Antarctic ice shelves to swell, in support of previous findings based on frequency-domain analysis.

Figures (9)

• Figure 1: Schematic (not to scale) of the equilibrium geometry.
• Figure 2: Heatmaps showing ratios of (a) maximum strain due to extensional waves to maximum strain due to flexural waves in the frequency domain, $\max\vert\varepsilon_{\text{ext}}\vert\,/\,\max\vert\varepsilon_{\text{flex}}\vert$, and (b) phase speed of the extensional wave to that of the flexural wave, $c_{\text{ext}}\,/\,c_{\text{flex}}$, versus wave period, $T$, and ice shelf thickness, $D$.
• Figure 3: Snapshots of the strain field, $\epsilon(x,z,t)$ (\ref{['eq:strain']}), in a $D=200$ m-thick ice shelf, forced by a Gaussian incident wave packet (\ref{['Eq:packet']}) with a peak period $T_{\text{peak}} = 15$ s and width $\sigma = 2 \times 10^{-3}$ m$^{-1}$. The free surface elevation of the incident packet, $\eta{(x,t)}=\eta_{\text{inc}}{(x,t)}$, is shown in the open ocean (black curve), and the displacement potential is shown in the sub-shelf water cavity, $\Phi(x,z,t)$ (\ref{['eqs:SMA_3']}b) (scaled by a factor $10^{-3}$ to match the colorbar for strain). The snapshots are at times (a) $t = 20$ s, (b) $100$ s, and (c) $200$ s.
• Figure 4: Snapshots of the strain profile along the bottom of the ice shelf (normalised by its maximum value at that instant of time; Eq. \ref{['eqs:strain_norm']}), $\tilde{\epsilon}(x,t)$ (blue curves), for the case shown in Fig. \ref{['fig:Strain_field']}. The corresponding contributions to the strain due to flexural waves, $\tilde{\epsilon}_{\text{flex}}(x,t)$ (red), and extensional waves, $\tilde{\epsilon}_{\text{ext}}(x,t)$ (green), are superimposed.
• Figure 5: (a) The strain profile relative to the maximum strain in the corresponding frequency-domain problem, $\hat{\epsilon}(x,t)$ (\ref{['eqs:strains_relative_1']}b), at the time when the maximum strain magnitude is attained, $t=t_{\max}$ (\ref{['eq:tmax']}) (blue curve), superimposed on the frequency-domain strain profile, $\hat{\varepsilon}(x)$ (\ref{['eqs:strains_relative_1']}a; black). (b) Corresponding strain profiles due to flexural waves, $\hat{\epsilon}_{\text{flex}}(x,t)$ (\ref{['eqs:strains_relative_2']}a; red), and extensional waves, $\hat{\epsilon}_{\text{ext}}(x,t)$ (\ref{['eqs:strains_relative_2']}b; green). (c) Time series of the strains due to flexural and extensional waves at the location where the maximum strain magnitude is attained, $x=x_{\max}$ (\ref{['eq:xmax']}).