360 research outputs found
A Comparison of Two Shallow Water Models with Non-Conforming Adaptive Grids: classical tests
In an effort to study the applicability of adaptive mesh refinement (AMR)
techniques to atmospheric models an interpolation-based spectral element
shallow water model on a cubed-sphere grid is compared to a block-structured
finite volume method in latitude-longitude geometry. Both models utilize a
non-conforming adaptation approach which doubles the resolution at fine-coarse
mesh interfaces. The underlying AMR libraries are quad-tree based and ensure
that neighboring regions can only differ by one refinement level.
The models are compared via selected test cases from a standard test suite
for the shallow water equations. They include the advection of a cosine bell, a
steady-state geostrophic flow, a flow over an idealized mountain and a
Rossby-Haurwitz wave. Both static and dynamics adaptations are evaluated which
reveal the strengths and weaknesses of the AMR techniques. Overall, the AMR
simulations show that both models successfully place static and dynamic
adaptations in local regions without requiring a fine grid in the global
domain. The adaptive grids reliably track features of interests without visible
distortions or noise at mesh interfaces. Simple threshold adaptation criteria
for the geopotential height and the relative vorticity are assessed.Comment: 25 pages, 11 figures, preprin
Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere
We describe a compatible finite element discretisation for the shallow water
equations on the rotating sphere, concentrating on integrating consistent
upwind stabilisation into the framework. Although the prognostic variables are
velocity and layer depth, the discretisation has a diagnostic potential
vorticity that satisfies a stable upwinded advection equation through a
Taylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at
the gridscale whilst retaining optimal order consistency. We also use upwind
discontinuous Galerkin schemes for the transport of layer depth. These
transport schemes are incorporated into a semi-implicit formulation that is
facilitated by a hybridisation method for solving the resulting mixed Helmholtz
equation. We illustrate our discretisation with some standard rotating sphere
test problems.Comment: accepted versio
Symmetric Equations on the Surface of a Sphere as Used by Model GISS:IB
Standard vector calculus formulas of Cartesian three space are projected onto the surface of a sphere. This produces symmetric equations with three nonindependent horizontal velocity components. Each orthogonal axis has a velocity component that rotates around its axis (eastward velocity rotates around the northsouth axis) and a specific angular momentum component that is the product of the velocity component multiplied by the cosine of axis latitude. Angular momentum components align with the fixed axes and simplify several formulas, whereas the rotating velocity components are not orthogonal and vary with location. Three symmetric coordinates allow vector resolution and calculus operations continuously over the whole spherical surface, which is not possible with only two coordinates. The symmetric equations are applied to one-layer shallow water models on cubed-sphere and icosahedral grids, the latter being computationally simple and applicable to an ocean domain. Model results are presented for three different initial conditions and five different resolutions
A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: comparison of hexagonal–icosahedral and cubed-sphere grids
A new algorithm is presented for the solution of the shallow water
equations on quasi-uniform spherical grids. It combines a mimetic
finite volume spatial discretization with a Crank–Nicolson time
discretization of fast waves and an accurate and conservative
forward-in-time advection scheme for mass and potential vorticity
(PV). The algorithm is implemented and tested on two families of
grids: hexagonal–icosahedral Voronoi grids, and modified equiangular
cubed-sphere grids.
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Results of a variety of tests are presented, including convergence
of the discrete scalar Laplacian and Coriolis operators, advection,
solid body rotation, flow over an isolated mountain, and
a barotropically unstable jet. The results confirm a number of
desirable properties for which the scheme was designed: exact mass
conservation, very good available energy and potential enstrophy
conservation, consistent mass, PV and tracer transport, and good
preservation of balance including vanishing ∇ × ∇,
steady geostrophic modes, and accurate PV advection. The scheme is
stable for large wave Courant numbers and advective Courant numbers
up to about 1.
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In the most idealized tests the overall accuracy of the scheme
appears to be limited by the accuracy of the Coriolis and other
mimetic spatial operators, particularly on the cubed-sphere grid.
On the hexagonal grid there is no evidence for damaging effects of
computational Rossby modes, despite attempts to force them
explicitly
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