Similitude of ice dynamics against scaling of geometry and physical parameters

Abstract

Abstract. The concept of similitude is commonly employed in the fields of fluid dynamics and engineering but rarely used in cryospheric research. Here we apply this method to the problem of ice flow to examine the dynamic similitude of isothermal ice sheets in shallow-shelf approximation against the scaling of their geometry and physical parameters. Carrying out a dimensional analysis of the stress balance we obtain dimensionless numbers that characterize the flow. Requiring that these numbers remain the same under scaling we obtain conditions that relate the geometric scaling factors, the parameters for the ice softness, surface mass balance and basal friction as well as the ice-sheet intrinsic response time to each other. We demonstrate that these scaling laws are the same for both the (two-dimensional) flow-line case and the three-dimensional case. The theoretically predicted ice-sheet scaling behavior agrees with results from numerical simulations that we conduct in flow-line and three-dimensional conceptual setups. We further investigate analytically the implications of geometric scaling of ice sheets for their response time. With this study we provide a framework which, under several assumptions, allows for a fundamental comparison of the ice-dynamic behavior across different scales. It proves to be useful in the design of conceptual numerical model setups and could also be helpful for designing laboratory glacier experiments. The concept might also be applied to real-world systems, e.g., to examine the response times of glaciers, ice streams or ice sheets to climatic perturbations.

Scaling of instability time-scales of Antarctic outlet glaciers based on one-dimensional similitude analysis

“…The derivation of the scaling laws from these two equations is presented in full detail in Feldmann and Levermann (2016) and thus shall be outlined only very briefly here: in their dimensionless form the SSA and the equation of mass conservation together have three independent numbers, analogous to the Reynolds number in the Navier-Stokes equation (Reynolds, 1883).…”

“…(3), an initially stable situation with stronger accumulation but initially thinner ice at the grounding line, for example, results in a faster response in case of destabilization. As detailed in (Feldmann and Levermann, 2016), the above scaling laws (Eqs. 2 and 3) are consistent with analytic solutions of the ice-dynamic equations (Schoof, 2007).…”

“…More precisely, the scaling condition comes out of the non-dimensionalized frictional term of the SSA (see Eqs. 9 and 11 in Feldmann and Levermann, 2016). In its general form it reads…”

“…The friction exponent m stems from the Weertman-type sliding law that enters the SSA (see Eq. 3 in Feldmann and Levermann, 2016). The friction data applied in this study is obtained from an inversion model that assumes a linear relation between basal friction and ice velocity (m = 1) with spatially varying proportionality constant C (Morlighem et al, 2013).…”

“…To this end the presented method combines observational and model data with a similarity analysis of the governing equations for shallow ice-stream flow. This similarity analysis has been carried out in previous work (Feldmann and Levermann, 2016), yielding scaling laws that determine how the geometry, the time scale and the involved physical parameters for ice softness, surface mass balance and basal friction have to relate in order to satisfy similitude between different ice sheets. The approach is an analogy to, e.g., the derivation of the Reynold's number (Reynolds, 1883) from the Navier-Stokes equation (Kundu et al, 2012), which provides a scaling law for a fluid's characteristic dimension, velocity and viscosity that assures similar flow patterns.…”