Journal cover Journal topic
Geoscientific Model Development An interactive open-access journal of the European Geosciences Union
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
On computation of Hough functions
Houjun Wang1,2, John P. Boyd3, and Rashid A. Akmaev2 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.

Citation: Wang, H., Boyd, J. P., and Akmaev, R. A.: On computation of Hough functions, Geosci. Model Dev., 9, 1477-1488,, 2016.
Publications Copernicus
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...