Leighton's bounds for Sturm-Liouville eigenvalues with eigenvalue parameter in the boundary conditions

2021

J. Glaciol.

This paper examines the effect of basal topography and strength on the grounding-line position, flux and stability of rapidly-sliding ice streams. It does so by supposing that the buoyancy of the ice stream is small, and of the same order as the longitudinal stress gradient. Making this scaling assumption makes the role of the basal gradient and accumulation rate explicit in the lowest order expression for the ice flux at the grounding line and also provides the transcendental equation for the grounding-line position. It also introduces into the stability condition terms in the basal curvature and accumulation-rate gradient. These expressions revert to well-established expressions in circumstances in which the thickness gradient is large at the grounding line, a result which is shown to be the consequence of the non-linearity of the flow. The behaviour of the grounding-line flux is illustrated for a range of bed topographies and strengths. We show that, when bed topography at a horizontal scale of several tens of ice thicknesses is present, the grounding-line flux and stability have more complex dependencies on bed gradient than that associated with the ‘marine ice-sheet instability hypothesis’, and that unstable grounding-line positions can occur on prograde beds as well as stable positions on retrograde beds.

Section: Appendix a Schoof's Boundary Layersupporting

confidence: 77%

Section: Appendix a Schoof's Boundary Layermentioning

confidence: 99%

Bed topography and marine ice-sheet stability

2021

J. Glaciol.

“…The evolution of the grounding line in the corresponding linearized perturbation problem (Supplementary Methods 2 ) is described by , where Λ is an eigenvalue. A stability condition for the case of relies on properties of the eigenfunctions of the perturbation problem 27 and is determined by the sign of the largest eigenvalue—either the fastest growing, in the case of an unstable grounding line position, or the slowest decaying in the case of a stable grounding line position 20 . In the case of no inferences about either the sign or magnitude of eigenvalues can be made; and the perturbation eigenvalue problem has to be solved explicitly to determine the largest eigenvalue.…”

Section: Resultsmentioning

confidence: 99%

No general stability conditions for marine ice-sheet grounding lines in the presence of feedbacks

2022

Nat Commun

The “marine ice-sheet instability” hypothesis continues to be used to interpret the observed mass loss from the Antarctic and Greenland ice sheets. This hypothesis has been developed for conditions that do not account for feedbacks between ice sheets and environmental conditions. However, snow accumulation and the ice-sheet surface melting depend on the surface temperature, which is a strong function of elevation. Consequently, there is a feedback between precipitation, atmospheric surface temperature and ice-sheet surface elevation. Here, we investigate stability conditions of a marine-based ice sheet in the presence of such a feedback. Our results show that no general stability condition similar to one associated with the “marine ice-sheet instability” hypothesis can be determined. Stability of individual configurations can be established only on a case-by-case basis. These results apply to a wide range of feedbacks between marine ice sheets and atmosphere, ocean and lithosphere.

“…The sign of the first term in square brackets on the right-hand side is positive when A 2 is positive. This is due to the fact that, the eigenfunction h corresponding to the largest value of Λ has no zeros in accordance with Theorem 1 of Linden (1991), provided A 2 > 0; and consequently the term xc 0 hdx ′ / h is positive. The sign of the denominator ĥx − h cx is known for a given calving rule (17-19) and a particular steady-state configuration (12).…”

Section: Appendix a A Linear Stability Analysismentioning

confidence: 85%

Marine outlet glacier dynamics, steady states and steady-state stability

2022

J. Glaciol.

Laterally confined marine outlet glaciers exhibit a diverse range of behaviours. This study investigates time-evolving and steady configurations of such glaciers. Using simplified analytic models, it determines conditions for steady states, their stability and expressions for the rate of the calving-front migration for three widely used calving rules. It also investigates the effects of ice mélange when it is present. The results show that ice flux at the terminus is an implicit function of ice thickness that depends on the glacier geometric and dynamic parameters. As a consequence, stability of steady-state configurations is determined by a complex combination of these parameters, specifics of the calving rule and the details of mélange stress conditions. The derived expressions of the rate of terminus migration suggest a non-linear feedback between the migration rate and the calving-front position. A close agreement between the obtained analytic expressions and numerical simulations suggests that these expressions can be used to gain insights into the observed behaviour of the glaciers and also to use observations to improve understanding of calving conditions.