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Volume 9, issue 4 | Copyright
Geosci. Model Dev., 9, 1477-1488, 2016
© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.

Development and technical paper 20 Apr 2016

Development and technical paper | 20 Apr 2016

On computation of Hough functions

Houjun Wang1,2, John P. Boyd3, and Rashid A. Akmaev2 Houjun Wang et al.
  • 1CIRES, University of Colorado Boulder, Boulder, Colorado, USA
  • 2Space Weather Prediction Center, NOAA, Boulder, Colorado, USA
  • 3Department of Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, Michigan, USA

Abstract. Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the commonly used unnormalized associated Legendre polynomial expansion method. Comparable results are obtained using both methods. For the first method we note some details on numerical implementation. The Chebyshev collocation method was first used for the Laplace tidal problem by Boyd (1976) and is relatively easy to use. A compact MATLAB code is provided for this method. We also illustrate the importance and effect of including a parity factor in Chebyshev polynomial expansions for modes with odd zonal wave numbers.

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Short summary
We briefly survey numerical methods for computing eigenvalues and eigenfunctions for the Laplace tidal equation. In particular we compare two methods that have numerical or conceptual advantages over the most commonly used methods. MATLAB codes are provided to facilitate their use. Researchers interested in atmospheric tidal analysis or in numerical methods for accurately computing eigenvalues of differential operators may find the paper helpful.
We briefly survey numerical methods for computing eigenvalues and eigenfunctions for the Laplace...