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We present new numerical methods for the shallow water equations on a
sphere in spherical coordinates.
In our implementation, the equations are discretized
in time with the two-level
semi-Lagrangian semi-implicit (SLSI) method, and in space on a staggered
grid with the quadratic spline Galerkin and the optimal
quadratic spline collocation methods.
When discretized on a uniform spatial grid, the solutions
are shown through numerical experiments to be fourth-order
in space locally at the nodes and midpoints of the spatial grids,
and third-order globally.
We also show that, when applied to a simplified version of the shallow
water equations, each of our algorithms yields a neutrally stable
solution for the
meteorologically significant Rossby waves.
Moreover, we demonstrate that the Helmholtz equation associated with the
shallow water
equations should be derived algebraically rather than analytically in
order for the algorithms
to be stable with respect to the Rossby waves.
These results are verified numerically using Boyd's equatorial wave equations
with initial conditions to generate a soliton.
Simulation of a Rossby Soliton
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We then analyze the performance of our methods on various staggered grids:
the A-, B-, and C-grids.
From an eigenvalue analysis of our simplified version of the shallow
water equations, we conclude that, when discretized on the C-grid,
our algorithms faithfully capture the group velocity of inertia-gravity waves.
Our analysis suggests that neither the A- nor B-grids will produce
such good results.
Our theoretical results are supported by numerical experiments, in which
we discretize Boyd's equatorial wave equations using different staggered
grids and set the initial conditions for the problem to generate
gravitation modes instead of a soliton.
With both the A- and B-grids, some waves are observed to travel in the
wrong direction, whereas, with the C-grid, gravity waves of all wavelengths
propagate in the correct direction.
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