A moving-point approach to model shallow ice sheets: a study case with radially symmetrical ice sheets

Predicting the evolution of ice sheets requires numerical models able to accurately track the migration of ice sheet continental margins or grounding lines. We introduce a physically based moving-point approach for the flow of ice sheets based on the conservation of local masses. This allows the ice sheet margins to be tracked explicitly. Our approach is also well suited to capture waiting-time behaviour efficiently. A finite-difference moving-point scheme is derived and applied in a simplified context (continental radially symmetrical shallow ice approximation). The scheme, which is inexpensive, is verified by comparing the results with steady states obtained from an analytic solution and with exact moving-margin transient solutions. In both cases the scheme is able to track the position of the ice sheet margin with high accuracy

Data assimilation in glaciology

International audienceIn this short paper, we will give one example of an inverse problem in glaciology. This problem is fairly simple to state: how to infer a climatic scenario (i.e. how to reconstruct past polar temperature) from ice volume records? The idea of this work is to explore the ability of the adjoint method to solve the inverse problem of reconstructing past temperature given all available observations. We start here with a simplified ice-sheet model and perform twin experiments

An ETKF approach for initial state and parameter estimation in ice sheet modelling

International audienceEstimating the contribution of Antarctica and Greenland to sea-level rise is a hot topic in glaciology. Good estimates rely on our ability to run a precisely calibrated ice sheet evolution model starting from a reliable initial state. Data assimilation aims to provide an answer to this problem by combining the model equations with observations. In this paper we aim to study a state-of-the-art ensemble Kalman filter (ETKF) to address this problem. This method is implemented and validated in the twin experiments framework for a shallow ice flowline model of ice dynamics. The results are very encouraging, as they show a good convergence of the ETKF (with localisation and inflation), even for small-sized ensembles

Data assimilation for moving mesh methods with an application to ice sheet modelling

We develop data assimilation techniques for nonlinear dynamical systems modelled by moving mesh methods. Such techniques are valuable for explicitly tracking interfaces and boundaries in evolving systems. The unique aspect of these assimilation techniques is that both the states of the system and the positions of the mesh points are updated simultaneously using physical observations. Covariances between states and mesh points are generated either by a correlation structure function in a variational context or by ensemble methods. The application of the techniques is demonstrated on a one-dimensional model of a grounded shallow ice sheet. It is shown, using observations of surface elevation and/or surface ice velocities, that the techniques predict the evolution of the ice sheet margin and the ice thickness accurately and efficiently. This approach also allows the straightforward assimilation of observations of the position of the ice sheet margin

Complex solutions of Monge-Amp\`ere equations

We describe a method to reduce partial differential equations of Monge-Amp\`ere type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Amp\`ere equations and the Plebanski equations.Comment: 16 pages, 5 tables To appear in Journal of Geometry and Physic

A multi-sourced assessment of the spatiotemporal dynamics of soil moisture in the MARINE flash flood model

The MARINE (Model of Anticipation of Runoff and INundations for Extreme events) hydrological model is a distributed model dedicated to flash flood simulation. Recent developments of the MARINE model are explored in this work. On one hand, transfers of water through the subsurface, formerly relying on water height, now take place in a homogeneous soil column based on the soil saturation degree (SSF model). On the other hand, the soil column is divided into two layers, which represent, respectively, the upper soil layer and the deep weathered rocks (SSF–DWF model). The aim of the present work is to assess the accuracy of these new representations for the simulation of soil moisture during flash flood events. An exploration of the various products available in the literature for soil moisture estimation is performed. The efficiency of the models for soil saturation degree simulation is estimated with respect to several products either at the local scale or spatially distributed: (i) the gridded soil moisture product provided by the operational modeling chain SAFRAN-ISBA-MODCOU; (ii) the gridded soil moisture product provided by the LDAS-Monde assimilation chain, which is based on the ISBA-A-gs land surface model and assimilating satellite derived data; (iii) the upper soil water content hourly measurements taken from the SMOSMANIA observation network; and (iv) the Soil Water Index provided by the Copernicus Global Land Service (CGLS), which is derived from Sentinel-1 C-SAR and ASCAT satellite data. The case study is performed over two French Mediterranean catchments impacted by flash flood events over the 2017–2019 period. The local comparison of the MARINE outputs with the SMOSMANIA measurements, as well as the comparison at the basin scale of the MARINE outputs with the gridded LDAS-Monde and CGLS data, lead to the following conclusion: both the dynamics and the amplitudes of the soil saturation degree simulated with the SSF and SSF–DWF models are better correlated with both the SMOSMANIA measurements and the LDAS-Monde data than the outputs of the base model. Finally, the soil saturation degree simulated by the two-layers model for the deep layer is compared to the soil saturation degree provided by the LDAS-Monde product at corresponding depths. In conclusion, the developments presented for the representation of subsurface flow in the MARINE model enhance the soil saturation degree simulation during flash floods with respect to both gridded data and local soil moisture measurements

The PDEs of biorthogonal polynomials arising in the two-matrix model

The two-matrix model can be solved by introducing bi-orthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of bi-orthogonal polynomials (called "windows") satisfy polynomial ODEs as well as deformation equations (PDEs) and finite difference equations (Delta-E) which are all Frobenius compatible and define discrete and continuous isomonodromic deformations for the irregular ODE, as shown in previous works of ours. In the one matrix model an explicit and concise expression for the coefficients of these systems is known and it allows to relate the partition function with the isomonodromic tau-function of the overdetermined system. Here, we provide the generalization of those expressions to the case of bi-orthogonal polynomials, which enables us to compute the determinant of the fundamental solution of the overdetermined system of ODE+PDEs+Delta-E.Comment: 20 pages v1 18 Nov 2003; v2 9 Jan 2004: trivial Latex mistake correcte