Shallow Water Equations on a Sphere
Anita Layton
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An accurate and timely forecast of weather elements, including the
the behavior of the atmosphere, ocean water and sea ice, is of great
importance to both public safety and to the world economy.
The accuracy of weather prediction depends on many factors, including
the accuracy of the knowledge of the state of the atmosphere at
the initial time, the numerical methods applied, and the resolution used
in these methods.
Weather prediction computations are known to be very time-consuming.
Therefore, there is a lot of interest among the scientific community in
studying accurate and efficient methods for weather prediction.
One way to achieve high accuracy in weather prediction computations is to
consider high-order discretization methods.
The shallow water equations (SWEs), which describe the inviscid flow of
a thin layer of fluid in two dimensions,
have been used for many years by the
atmospheric modeling community as a vehicle for testing promising numerical
methods for solving atmospheric and oceanic problems.
Because the Earth is approximately spherical, most global atmospheric
models in use today are based on spherical coordinates.
In most global meteorological applications, spatial discretization schemes
are based on spectral transform methods (STM)
or low-order finite difference/finite element methods.
The STM yields high-order solutions and gives rise to elliptic
equations that are computationally inexpensive to solve.
However, the STM also has some disadvantages.
Provided that an optimal solver is applied for the solution of the
linear system arising from the Helmholtz problem, the computational
cost of finite difference and finite element methods
applied to the shallow water equations (SWEs) on the sphere
increases quadratically with the number of gridpoints in one dimension
(i.e., O(N^2), where N is the number of spatial subintervals
in one dimension). However, the cost of performing spectral transforms
increases more rapidly. In the case of Fourier transforms
in the longitudinal direction,
fast Fourier Transforms (FFTs) may be used and their computational cost
grows as O(N^2 log N). An efficient method for performing
Legendre transforms, analogous to FFTs, has not yet been developed. Thus,
the Legendre transforms in the the latitudinal direction are often
performed by summation and their costs escalate at a rate of
O(N^3).
Moreover, the spectral method is formally equivalent to a
least squares approximation that minimizes the mean square error over the
global domain. This implies that the size of the error is likely to be the
same everywhere. This may be a serious disadvantage in more comprehensive
atmospheric models for a field, such as water vapor, for which the average
value varies greatly over the globe. In the case of water vapor, for example,
a small absolute error may be insignificant in equatorial regions,
but it may completely alter the character of the field in polar regions.
Thus, there is interest in the atmospheric community in developing
alternative high-order numerical methods.
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Last updated: April 24, 2004.
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