Application of a two-step approach for mapping ice thickness to various glacier types on Svalbard

The basal topography is largely unknown beneath most glaciers and ice caps, and many attempts have been made to estimate a thickness field from other more accessible information at the surface. Here, we present a two-step reconstruction approach for ice thickness that solves mass conservation over single or several connected drainage basins. The approach is applied to a variety of test geometries with abundant thickness measurements including marine- and land-terminating glaciers as well as a 2400 km2 ice cap on Svalbard. The input requirements are kept to a minimum for the first step. In this step, a geometrically controlled, non-local flux solution is converted into thickness values relying on the shallow ice approximation (SIA). In a second step, the thickness field is updated along fast-flowing glacier trunks on the basis of velocity observations. Both steps account for available thickness measurements. Each thickness field is presented together with an error-estimate map based on a formal propagation of input uncertainties. These error estimates point out that the thickness field is least constrained near ice divides or in other stagnant areas. Withholding a share of the thickness measurements, error estimates tend to overestimate mismatch values in a median sense. We also have to accept an aggregate uncertainty of at least 25 % in the reconstructed thickness field for glaciers with very sparse or no observations. For Vestfonna ice cap (VIC), a previous ice volume estimate based on the same measurement record as used here has to be corrected upward by 22 %. We also find that a 13 % area fraction of the ice cap is in fact grounded below sea level. The former 5 % estimate from a direct measurement interpolation exceeds an aggregate maximum range of 6–23 % as inferred from the error estimates here.

Application of a two-step approach for mapping ice thickness to various glacier types on Svalbard

The basal topography is largely unknown beneath most glaciers and ice caps and many attempts have been made to estimate a thickness field from other more accessible information at the surface. Here, we present a two-step reconstruction approach for ice thickness that solves mass conservation over single or several connected drainage basins. The approach performs well for a variety of test geometries with abundant thickness measurements including marine- and land-terminating glaciers as well as a 2400 km2 ice cap on Svalbard. Input requirements for the first step are comparable to other approaches that have already been applied world-wide. In the first step, a geometrically controlled, non-local flux solution is converted into thickness values relying on the shallow ice approximation. In a second step, the thickness reconstruction is improved along fast-flowing glacier trunks on the basis of velocity observations. In both steps, thickness measurements are assimilated as internal boundary conditions. Each thickness field is presented together with a map of error estimates which stem from a formal propagation of input uncertainties. These estimates point out that the thickness field is least constrained near ice divides or in other stagnant areas. The error-estimate map also highlights key regions for future thickness surveys as well as a preference for across-flow acquisition. Withholding parts of the thickness measurements indicates that error estimates show a tendency to overestimate actual mismatch values. For very sparse or non-existent thickness information, our reconstruction approach indicates that we have to accept an average uncertainty of at least 25 % in the reconstructed thickness field. For Vestfonna, previous ice volume estimates have to be corrected upward by 22 %. We also find that a 12 % area fraction of the ice cap are in fact grounded below sea-level as compared to the previous 5 %-estimate.

Free radially expanding liquid sheet in air: time- and space-resolved measurement of the thickness field

The collision of a liquid drop against a small target results in the formation of a thin liquid sheet that extends radially until it reaches a maximum diameter. The subsequent retraction is due to the air–liquid surface tension. We have used a time- and space-resolved technique to measure the thickness field of this class of liquid sheet, based on the grey-level measurement of the image of a dyed liquid sheet recorded using a high-speed camera. This method enables a precise measurement of the thickness in the range $10{-}450~{\rm\mu}\text{m}$, with a temporal resolution equal to that of the camera. We have measured the evolution with time since impact, $t$, and radial position, $r$, of the thickness, $h(r,t)$, for various drop volumes and impact velocities. Two asymptotic regimes for the expansion of the sheet are evidenced. The scalings of the thickness with $t$ and $r$ measured in the two regimes are those that were predicted by Rozhkov et al. (Proc. R. Soc. Lond. A, vol. 460, 2004, pp. 2681–2704) for the short-time regime and Villermaux and Bossa (J. Fluid Mech., vol. 668, 2011, pp. 412–435) for the long-time regime, but never experimentally measured before. Interestingly, our experimental data also provide evidence for the existence of a maximum of the film thickness $h_{max}(r)$ at a radial position $r_{h_{max}}(t)$ corresponding to the cross-over of these two asymptotic regimes. The maximum moves with a constant velocity of the order of the drop impact velocity, as expected theoretically. Thanks to our visualization technique, we also provide evidence of an azimuthal thickness modulation of the liquid sheets.

Analysis of Mode I Conducting Crack in Piezo-Electro-Magneto-Elastic Layer

Within the theory of linear magnetoelectroelasticity, the fracture analysis of a magneto - electrically dielectric crack embedded in a magnetoelectroelastic layer is investigated. The prescribed displacement, electric potential and magnetic potential boundary conditions on the layer surfaces are adopted. Applying the Hankel transform technique, the boundary - value problem is reduced to solving three coupling Fredholm integral equations of second kind. These equations are solved exactly. The corresponding semi - permeable crack - face magnetoelectric boundary conditions are adopted and the electric displacement and magnetic induction of crack interior are obtained explicitly. This field inside the crack is dependent on the material properties, applied loadings, the dielectric permittivity and magnetic permeability of crack interior, and the ratio of the crack length and the layer thickness. Field intensity factors are obtained as explicit expressions.