Supplement of Mechanics and dynamics of pinning points on the Shirase Coast, West Antarctica

Abstract. Ice rises and rumples, sites of localised ice-shelf grounding, modify ice-shelf flow by generating lateral and basal shear stresses, upstream compression, and downstream tension. Studies of pinning points typically quantify this role indirectly, through related metrics such as a buttressing number. Here, we quantify the dynamic effects of pinning points directly, by comparing model-simulated stress states in the Ross Ice Shelf (RIS) with and without a specific set of pinning points located downstream of the MacAyeal and Bindschadler ice streams (MacIS and BIS, respectively). Because ice properties are only known indirectly, the experiment is repeated with different realisations of the ice softness. While longitudinal stretching, and thus ice velocity, is smaller with the pinning points, flow resistance generated by other grounded features is also smaller. Conversely, flow resistance generated by other grounded features increases when the pinning points are absent, providing a non-local control on the net effect of the pinning points on ice-shelf flow. We find that an ice stream located directly upstream of the pinning points, MacIS, is less responsive to their removal than the obliquely oriented BIS. This response is due to zones of locally higher basal drag acting on MacIS, which may itself be a consequence of the coupled ice-shelf and ice-stream response to the pinning points. We also find that inversion of present-day flow and thickness for basal friction and ice softness, without feature-specific, a posteriori adjustment, leads to the incorrect representation of ice rumple morphology and an incorrect boundary condition at the ice base. Viewed from the perspective of change detection, we find that, following pinning point removal, the ice shelf undergoes an adjustment to a new steady state that involves an initial increase in ice speeds across the eastern ice shelf, followed by decaying flow speeds, as mass flux reduces thickness gradients in some areas and increases thickness gradients in others. Increases in ice-stream flow speeds persist with no further adjustment, even without sustained grounding-line retreat. Where pinning point effects are important, model tuning that respects their morphology is necessary to represent the system as a whole and inform interpretations of observed change.


1 Model set-up and comparison between observations and model output 1.1 Physical constants Table S1. Physical constants used in the Ice-sheet and Sea-level System Model (ISSM). Glen's flow law exponent n 3

Improvements to the model representation of pinning points
Some minor adjustments are made to model boundaries before relaxation. The friction coefficient assigned to Crary Ice Rise and Steershead Ice Rise is increased to ∞ to ensure zero velocity, characteristic of ice rises. Model bathymetry beneath the Ross 5 Ice Shelf (RIS) is adjusted to prevent the formation of new ice rumples during relaxation. This adjustment is necessary because the Bedmap2 sub-ice shelf bathymetry used to create the model geometry is interpolated from limited resolution (55 km) point measurements onto a 1 km grid (Fretwell et al., 2013). Predictably, the depth of the water column beneath the RIS is incorrect in some areas and this poses particular problems where the seafloor is unrealistically shallow. Water column depths of less than 50 m near the Shirase Coast Ice Rumples (SCIR) and Steershead Ice Rise (SIR) result in spurious grounding and the formation 10 of new ice rumples, as well an increase in the area of existing pinning points.
Unrealistic grounding is prevented by excavating a 500 m thick layer from the bed elevation beneath the RIS (Fig. S1).  Fürst et al. (2015) and Favier et al. (2016) who also noted inconsistencies in the Bedmap2 bathymetry that resulted in spurious regrounding when simulating ice shelf flow over pinning points.
The friction coefficient initially inferred for the SCIR (described in Section 2.2) is adjusted to achieve a more realistic  the friction coefficient after manual adjustment. this representation is incorrect. Downstream rumples provide basal drag and generate surface relief comparable to the upstream rumples in the complex (implying that basal drag is greater than zero).

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To examine sensitivity to basal friction, a range of different friction coefficient values (α = 0 to 800 s 1/2 m −1/2 ) were assigned to the ice rumple nodes before model relaxation. A friction coefficient value that reproduces a relaxed model geometry close to present-day ice velocity and thickness, and that produces a basal drag magnitude similar to the basal drag inferred in a force budget analysis (Still et al., 2019), is used in the model experiments (i.e. α = 200 s 1/2 m −1/2 ). The manual adjustment ensured that all of the individual ice rumples in the complex (and the associated velocity gradients and resistive stresses) were 30 represented appropriately.

Performance of the inverse method
Reference model velocities are well matched to observed velocities (Table S2,   While the broad pattern of thickening upstream and thinning downstream of the SCIR is preserved, kilometre-scale features including wakes of thinner ice downstream from individual ice rumples are not reproduced (Fig. S5).
A realistic assessment of the flow-regulating effects of the SCIR requires a model ice-shelf thickness above buoyancy H ab that is consistent with the observed H ab from the Bedmap2 compilation. If the model H ab is too high, the flow-regulating 45 effects of the pinning points will be exaggerated, while if H ab is too low (ice rumples are too lightly grounded), the flowregulating effects will be underestimated. After model relaxation, H ab ranges from -6 m to 23 m with a mean H ab of 13 m ( Fig. S5c and d). Negative H ab values indicate that some initially grounded ice rumple nodes lost contact with the seafloor during model relaxation (e.g. the downstream western corner of rumple C).

Transverse stresses 50
Longitudinal tensile and compressive stresses in the across-flow direction are referred to as transverse stressesR tt . Positive and negativeR tt (indicating flow divergence and convergence) are plotted separately for clarity ( Fig. S6 and S7).

3 The force budget
The pinning point force budget quantifies the magnitude and direction of drag forces and net flow resistance generated by individual pinning points (MacAyeal et al., 1987;Still et al., 2019). Form drag F f is the glaciostatic contribution to the net flow 55 resistance associated with disturbance to the thickness field around an obstacle and is calculated on a contour Γ surrounding the pinning point where ρ i is the density of ice, g is acceleration due to gravity and H is the ice thickness. The contour Γ is divided into length elements dλ with normalsn. Equation (1) differs slightly from the F f equation used by MacAyeal et al. (1987) because the 60 total ISSM ice-column thickness H, in which a firn layer is not represented separately, is used here.
Dynamic drag F d is the viscous resistance associated with deformation around an obstacle and is calculated whereη is the effective depth-averaged viscosity and˙ ij is the strain rate tensor. The effective viscosity parameterη is whereB is the depth-averaged, inverse flow law rate factor and n = 3 is the flow law exponent. The effective strain rate˙ e is the second invariant of the strain rate tensoṙ 2 e = 1 2 (˙ 2 xx +˙ 2 yy +˙ 2 zz ) +˙ 2 xy +˙ 2 xz +˙ 2 yz .
The difference between the sum of the form drag and dynamic drag (F f + F d ) and the resistive force that would exist in the absence of any disturbance to ice shelf flow (the seawater pressure), is the effective resistance F e , the total reaction force 70 arising from contact between the pinning point and the ice shelf base where F w is the seawater pressure. The force budget components are computed for the SCIR and Roosevelt Island using both B inv andB u for the reference and perturbed models.

Roosevelt Island force budget 75
Modelled net resistive forces generated by Roosevelt Island increase in magnitude in response to removal of the SCIR. F f increases by 5%, F d increases by 6% and F e increases by 6% (F e = 37.8×10 12 N with SCIR, F e = 40.0×10 12 N without SCIR) ( Table 3). The directions of net F f , F d and F e remain unchanged (Fig. S9). Once the SCIR are removed, F e generated at the location of the former SCIR reduces to near-zero and F e generated by Roosevelt Island increases to balance the driving force. This result is consistent with the analysis of resistive stress distributions. In particular, lateral shear stressesR lt and 80 longitudinal compressive stresses −R ll generated by Roosevelt Island increase in response to removal of the SCIR.   The 'time' variable refers to the number of years after removal of the SCIR from the model domain.