scholarly journals Free–Surface Flow as a Variational Inequality (<i>evolve_glacier v1.1</i>): Numerical Aspects of a Glaciological Application

2020 ◽  
Author(s):  
Anna Wirbel ◽  
Alexander Helmut Jarosch
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Variational Inequality ◽  
Mass Balance ◽  
Ice Sheets ◽  
Computational Domain ◽  
Simulation Framework ◽  
Surface Evolution ◽  
Kinematic Boundary Condition ◽  
Finite Element Software

Abstract. Like any gravitationally driven flow that is not constrained at the upper surface, glaciers and ice sheets feature a free-surface, which becomes a free boundary problem within simulations. A kinematic boundary condition is often used to describe the evolution of this free-surface. However, in the case of glaciers and ice sheets, the naturally occurring constraint that the ice surface elevation (S) can not fall below the bed topography (B), (S-B > = 0) in combination with a non-zero mass balance rate complicates the matter substantially. We present an open-source numerical simulation framework to simulate the free-surface evolution of glaciers that directly incorporates this natural constraint. It is based on the finite element software package FEniCS solving the Stokes equations for ice flow and a suitable transport equation, i.e. 'kinematic boundary condition', for the free-surface evolution. The evolution of the free--surface is treated as a variational inequality, constrained by the bedrock underlying the glacier or the topography of the surrounding ground. To solve this problem, the 'constrained' non--linear problem solving capabilities of PETSc's SNES interface are used. As the constraint is considered in the solving process, this approach does not require any ad-hoc post-processing steps to enforce no--negativity of ice thickness as well as mass conservation. The simulation framework provides the possibility to partition the computational domain so that individual forms of the relevant equations can be solved for different subdomains all at once. In the presented setup, this is used to distinguish between glacierized and ice-free regions. The option to chose different time discretizations, spatial stabilisation schemes and adaptive mesh refinement make it a versatile tool for glaciological applications. We present a set of benchmark tests that highlight the simulation framework is able to reproduce the free-surface evolution of complex geometries under different conditions for which it is mass conserving and numerically stable. Real--world glacier examples demonstrate high resolution change in glacier geometry due to fully-resolved 3D velocities and spatially variable mass balance rate, whereby realistic glacier recession and advance states can be simulated. Additionally, we provide a thorough analysis of different spatial stabilisation techniques as well as time discretization methods. We discuss their applicability and suitability for different glaciological applications.

scholarly journals Inequality-constrained free-surface evolution in a full Stokes ice flow model (<i>evolve_glacier v1.1</i>)

2020 ◽  
Vol 13 (12) ◽  
pp. 6425-6445
Author(s):  
Anna Wirbel ◽  
Alexander Helmut Jarosch
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Mass Balance ◽  
Ice Sheets ◽  
Computational Domain ◽  
Simulation Framework ◽  
Ice Flow ◽  
Surface Evolution ◽  
Kinematic Boundary Condition ◽  
Non Linear

Abstract. Like any gravitationally driven flow that is not constrained at the upper surface, glaciers and ice sheets feature a free surface, which becomes a free-boundary problem within simulations. A kinematic boundary condition is often used to describe the evolution of this free surface. However, in the case of glaciers and ice sheets, the naturally occurring constraint that the ice surface elevation (S) cannot fall below the bed topography (B) (S-B≥0), in combination with a non-zero mass balance rate complicates the matter substantially. We present an open-source numerical simulation framework to simulate the free-surface evolution of glaciers that directly incorporates this natural constraint. It is based on the finite-element software package FEniCS solving the Stokes equations for ice flow and a suitable transport equation, i.e. “kinematic boundary condition”, for the free-surface evolution. The evolution of the free surface is treated as a variational inequality, constrained by the bedrock underlying the glacier or the topography of the surrounding ground. This problem is solved using a “reduced space” method, where a Newton line search is performed on a subset of the problem (Benson and Munson, 2006). Therefore, the “constrained” non-linear problem-solving capabilities of PETSc's (Portable, Extensible Toolkit for Scientific Computation, Balay et al., 2019) SNES (Scalable Non-linear Equations Solver) interface are used. As the constraint is considered in the solving process, this approach does not require any ad hoc post-processing steps to enforce non-negativity of ice thickness and corresponding mass conservation. The simulation framework provides the possibility to divide the computational domain into different subdomains so that individual forms of the relevant equations can be solved for different subdomains all at once. In the presented setup, this is used to distinguish between glacierised and ice-free regions. The option to chose different time discretisations, spatial stabilisation schemes and adaptive mesh refinement make it a versatile tool for glaciological applications. We present a set of benchmark tests that highlight that the simulation framework is able to reproduce the free-surface evolution of complex geometries under different conditions for which it is mass-conserving and numerically stable. Real-world glacier examples demonstrate high-resolution change in glacier geometry due to fully resolved 3D velocities and spatially variable mass balance rate, whereby realistic glacier recession and advance states can be simulated. Additionally, we provide a thorough analysis of different spatial stabilisation techniques as well as time discretisation methods. We discuss their applicability and suitability for different glaciological applications.

scholarly journals The Reconstruction of Former Ice Sheets and their Mass Balance Characteristics using a Non-Linearly Viscous Flow Model

1984 ◽  
Vol 30 (105) ◽  
pp. 140-152 ◽  
Author(s):  
G. S. Boulton ◽  
G. D. Smith ◽  
L. W. Morland
Keyword(s):  
Boundary Condition ◽  
Mass Balance ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Equilibrium Line Altitude ◽  
Isostatic Compensation ◽  
Geological Evidence ◽  
Wide Range ◽  
Terminal Zone ◽  
Subglacial Topography

AbstractA model of a non-linearly viscous ice sheet is used to investigate the influence of net mass-balance pattern, basal boundary condition, and subglacial topography on the size and shape of ice sheets. The aim is to enable geological evidence of the extent of former ice sheets to be used as indicators of palaeoclimate. A series of curves are presented showing the relationships between ice-sheet span, net mass balance, and equilibrium-line altitude (ELA) for zero and complete isostatic compensation. These are applicable to a very wide range of basal boundary conditions. The way in which they can be used to reconstruct net mass-balance gradients for former ice sheets is demonstrated. Changes in the basal boundary condition only have a strong influence on glacier span when they occur in the terminal zone. Ice-sheet expansion and contraction is not merely accompanied by changes in snow-line elevation, but also by changes in the net mass-balance gradient. The combinations of these required to cause ice-sheet expansion and contraction are analysed. A non-linearly viscous model for ice suggests that ice-sheet volume changes may not be a simple function of their change in areal extent.

scholarly journals A numerically exact non-reflecting boundary condition applied to the acoustic Helmholtz equation

Geophysics ◽  
2021 ◽  
pp. 1-42
Author(s):  
Wim A. Mulder
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Helmholtz Equation ◽  
Matrix Equation ◽  
Direct Method ◽  
Rectangular Domain ◽  
Translation Invariance ◽  
Computational Domain ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix

When modeling wave propagation, truncation of the computational domain to a manageable size requires non-reflecting boundaries. To construct such a boundary condition on one side of a rectangular domain for a finite-difference discretization of the acoustic wave equation in the frequency domain, the domain is extended on that one side to infinity. Constant extrapolation in the direction perpendicular to the boundary provides the material properties in the exterior. Domain decomposition can split the enlarged domain into the original one and its exterior. Because the boundary-value problem for the latter is translation-invariant, the boundary Green functions obey a quadratic matrix equation. Selection of the solvent that corresponds to the outgoing waves provides the input for the remaining problem in the interior. The result is a numerically exact non-reflecting boundary condition on one side of the domain. When two non-reflecting sides have a common corner, translation invariance is lost. Treating each side independently in combination with a classic absorbing condition in the other direction restores translation invariance and enables application of the method at the expense of numerical exactness. Solving the quadratic matrix equation with Newton's method turns out to be more costly than solving the Helmholtz equation and may select unwanted incoming waves. A proposed direct method has a much lower cost and selects the correct branch. A test on a 2-D acoustic marine seismic problem with a free surface at the top, a classic second-order Higdon condition at the bottom and numerically exact boundaries at the two lateral sides demonstrates the capability of the method. Numerically exact boundaries on each side, each computed independently with a free-surface or Higdon condition, provide even better results.

scholarly journals The Reconstruction of Former Ice Sheets and their Mass Balance Characteristics using a Non-Linearly Viscous Flow Model

1984 ◽  
Vol 30 (105) ◽  
pp. 140-152 ◽  
Author(s):  
G. S. Boulton ◽  
G. D. Smith ◽  
L. W. Morland
Keyword(s):  
Boundary Condition ◽  
Mass Balance ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Equilibrium Line Altitude ◽  
Isostatic Compensation ◽  
Geological Evidence ◽  
Wide Range ◽  
Terminal Zone ◽  
Subglacial Topography

AbstractA model of a non-linearly viscous ice sheet is used to investigate the influence of net mass-balance pattern, basal boundary condition, and subglacial topography on the size and shape of ice sheets. The aim is to enable geological evidence of the extent of former ice sheets to be used as indicators of palaeoclimate. A series of curves are presented showing the relationships between ice-sheet span, net mass balance, and equilibrium-line altitude (ELA) for zero and complete isostatic compensation. These are applicable to a very wide range of basal boundary conditions. The way in which they can be used to reconstruct net mass-balance gradients for former ice sheets is demonstrated. Changes in the basal boundary condition only have a strong influence on glacier span when they occur in the terminal zone. Ice-sheet expansion and contraction is not merely accompanied by changes in snow-line elevation, but also by changes in the net mass-balance gradient. The combinations of these required to cause ice-sheet expansion and contraction are analysed. A non-linearly viscous model for ice suggests that ice-sheet volume changes may not be a simple function of their change in areal extent.

Nonlinear Sloshing in Fixed and Vertically Excited Containers

2002 ◽  
Author(s):  
Jannette B. Frandsen ◽  
Alistair G. L. Borthwick
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Small Perturbation ◽  
Numerical Solutions ◽  
Vertical Direction ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Surface Flow ◽  
Nonlinear Behaviour ◽  
Computational Domain

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.

Interannual variations of the mass balance of the Antarctica and Greenland ice sheets from GRACE

2006 ◽  
Vol 53 (3) ◽  
pp. 198-208 ◽  
Author(s):  
G. Ramillien ◽  
A. Lombard ◽  
A. Cazenave ◽  
E.R. Ivins ◽  
M. Llubes ◽  
...  
Keyword(s):  
Mass Balance ◽  
Ice Sheets ◽  
Interannual Variations
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