Abstract
Abstract. Predictions of future mass loss from ice sheets are afflicted with uncertainty, caused, among others, by insufficient understanding of spatiotemporally variable processes at the inaccessible base of ice sheets for which few direct observations exist and of which basal friction is a prime example. Here, we present a general numerical framework for studying the relationship between bed and surface properties of ice sheets and glaciers. Specifically, we use an inverse modeling approach and the associated time-dependent adjoint equations, derived in the framework of a full Stokes model and a shallow-shelf/shelfy-stream approximation model, respectively, to determine the sensitivity of grounded ice sheet surface velocities and elevation to time-dependent perturbations in basal friction and basal topography. Analytical and numerical examples are presented showing the importance of including the time-dependent kinematic free surface equation for the elevation and its adjoint, in particular for observations of the elevation. A closed form of the analytical solutions to the adjoint equations is given for a two-dimensional vertical ice in steady state under the shallow-shelf approximation. There is a delay in time between a seasonal perturbation at the ice base and the observation of the change in elevation. A perturbation at the base in the topography has a direct effect in space at the surface above the perturbation, and a perturbation in the friction is propagated directly to the surface in time.
Highlights
Over the last decades, ice sheets and glaciers have experienced mass loss due to global warming, both in the polar regions and outside of Greenland and Antarctica (Farinotti et al, 2015; Mouginot et al, 2019; Pörtner et al, 2019; Rignot et al, 2019)
The flow of large bodies of ice is described with the help of the conservation laws of mass, momentum, and energy (Greve and Blatter, 2009), which together pose a system of nonlinear partial differential equations (PDEs) commonly referred to as the FS equations in glaciological applications
We show with the analytical solution in the FS model that the influence of ψ is negligible
Summary
Ice sheets and glaciers have experienced mass loss due to global warming, both in the polar regions and outside of Greenland and Antarctica (Farinotti et al, 2015; Mouginot et al, 2019; Pörtner et al, 2019; Rignot et al, 2019). Note that the same approach, here described for the case of the sliding law, can be used to determine other “inaccessibles”, such as optimal initial conditions for ice sheet modeling (Perego et al, 2014), the sensitivity of melt rates beneath ice shelves in response to ocean circulation (Heimbach and Losch, 2012), the geothermal heat flux at the ice base (Zhu et al, 2016), or to estimate basal topography beneath an ice sheet (Monnier and des Boscs, 2017; van Pelt et al, 2013) The latter is difficult to separate from the determination of the sliding properties (KyrkeSmith et al, 2018; Thorsteinsson et al, 2003), and has limitations related to the spatial resolution of surface data and/or measurement errors; see Gudmundsson (2003, 2008) and Gudmundsson and Raymond (2008). The analysis in this paper and the numerical experiments in Cheng and Lötstedt (2020) confirm the conclusion in Gudmundsson (2008) that the perturbations with a long wavelength and low frequency will propagate to the surface while those of a short wavelength and high frequency are damped
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