MPAS-Seaice (v1.0.0): Sea-ice dynamics on unstructured Voronoi meshes

We present MPAS-Seaice, a sea-ice model which uses the Model for Prediction Across Scales (MPAS) framework and Spherical Centroidal Voronoi Tessellation (SCVT) unstructured meshes. As well as SCVT meshes, MPAS-Seaice can run on the traditional quadrilateral grids used by sea-ice models such as CICE. The MPAS-Seaice velocity solver uses the Elastic-Viscous-Plastic (EVP) rheology, and the variational discretization of the internal stress divergence operator used by CICE, but adapted for the polygonal cells of MPAS meshes, or alternatively an integral (“weak”) formulation of the stress divergence operator. An incremental remapping advection scheme is used for mass and tracer transport. We validate these formulations with idealized test cases, both planar and on the sphere. The variational scheme displays lower errors than the weak formulation for the strain rate operator but higher errors for the stress divergence operator. The variational stress divergence operator displays increased errors around the pentagonal cells of a quasi-uniform mesh, which is ameliorated with an alternate formulation for the operator. MPAS-Seaice shares the sophisticated column physics and biogeochemistry of CICE, and when used with quadrilateral meshes can reproduce the results of CICE. We have used global simulations with realistic forcing to validate MPAS-Seaice against similar simulations with CICE and against observations. We find very similar results compared to CICE with differences explained by minor differences in implementation such as with interpolation between the primary and dual meshes at coastlines. We have assessed the computational performance of the model, which, because it is unstructured, runs 70 % as fast as CICE for a comparison quadrilateral simulation. The SCVT meshes used by MPAS-Seaice allow culling of equatorial model cells and flexibility in domain decomposition, improving model performance. MPAS-Seaice is the current sea-ice component of the Energy Exascale Earth System Model (E3SM).

Sobolev spaces with respect to a weighted Gaussian measure in infinite dimensions

Let [Formula: see text] be a separable Banach space endowed with a nondegenerate centered Gaussian measure [Formula: see text] and let [Formula: see text] be a positive function on [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text] and [Formula: see text]. In this paper, we introduce and study Sobolev spaces with respect to the weighted Gaussian measure [Formula: see text]. We obtain results regarding the divergence operator (i.e. the adjoint in [Formula: see text] of the gradient operator along the Cameron–Martin space) and the trace of Sobolev functions on hypersurfaces [Formula: see text], where [Formula: see text] is a suitable version of a Sobolev function.

The equation divu + 〈a,u〉 = f

We study the solutions [Formula: see text] to the equation [Formula: see text] where [Formula: see text] and [Formula: see text] are given. We significantly improve the existence results of [G. Csató and B. Dacorogna, A Dirichlet problem involving the divergence operator, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 829–848], where this equation has been considered for the first time. In particular, we prove the existence of a solution under essentially sharp regularity assumptions on the coefficients. The condition that we require on the vector field [Formula: see text] is necessary and sufficient. Finally, our results cover the whole scales of Sobolev and Hölder spaces.

Transport-Elliptic Estimates

This chapter solves the underdetermined, elliptic equation ∂ⱼQsuperscript jl = Usuperscript l and Qsuperscript jl = Qsuperscript lj (Equation 1069) in order to eliminate the error term in the parametrix. For the proof of the Main Lemma, estimates for Q and the material derivative as well as its spatial derivatives are derived. The chapter finds a solution to Equation (1069) with good transport properties by solving it via a Transport equation obtained by commuting the divergence operator with the material derivative. It concludes by showing the solutions, spatial derivative estimates, and material derivative estimates for the Transport-Elliptic equation, as well as cutting off the solution to the Transport-Elliptic equation.