Abstract
Abstract. We present the detailed construction of a manufactured analytical solution to time-dependent and steady-state isothermal full-Stokes ice sheet problems. The solutions are constructed for two-dimensional flowline and three-dimensional full-Stokes ice sheet models with variable viscosity. The construction is done by choosing for the specified ice surface and bed a velocity distribution that satisfies both mass conservation and the kinematic boundary conditions. Then a compensatory stress term in the conservation of momentum equations and their boundary conditions is calculated to make the chosen velocity distributions as well as the chosen pressure field into exact solutions. By substituting different ice surface and bed geometry formulas into the derived solution formulas, analytical solutions for different geometries can be constructed. The boundary conditions can be specified as essential Dirichlet conditions or as periodic boundary conditions. By changing a parameter value, the analytical solutions allow investigation of algorithms for a different range of aspect ratios as well as for different, frozen or sliding, basal conditions. The analytical solutions can also be used to estimate the numerical error of the method in the case when the effects of the boundary conditions are eliminated, that is, when the exact solution values are specified as inflow and outflow boundary conditions.
Highlights
Model verification is crucial in developing a numerical model
We present the detailed construction of a manufactured exact solution to time-dependent and steadystate isothermal full-Stokes ice sheet problems
The analytical solutions may help the modelers to estimate the numerical error in the case when the effect of the boundary conditions are eliminated, that is, when the exact solutions values are specified as inflow and outflow boundary conditions
Summary
The ice-sheet modeling community has been using two tools to verify models, comparison of numerically computed solutions to analytical solutions when possible, and intercomparison, that is, measuring differences between various models’ results on the sets of simplified geometry benchmark tests. Gudmundsson in (Gudmundsson, 2003) obtained the three-dimensional solution of the linearized zeroth-order problem for a linear viscous medium To use this solution for benchmarking numerical ice sheet models, the exact error estimate must be known (Raymond and Gudmundsson, 2005). The analytical solutions may help the modelers to estimate the numerical error in the case when the effect of the boundary conditions are eliminated, that is, when the exact solutions values are specified as inflow and outflow boundary conditions
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