On this page you will find:
1. Meridional structures of the Hough functions for the realistic background state of the mean zonal wind and for the background state at rest,
2. Horizontal structures of the Hough harmonics (Kelvin wave, mixed Rossbygravity wave, Rossby and inertiagravity waves) for different equivalent depths.
(a) Kelvin Waves
Horizontal structure of the n=0 eastwardpropagating inertiagravity mode on the sphere, Kelvin wave.
Geopotential height is scaled by the maximal height. Colours vary between 1 and +1 with positive/negative perturbations in red/blue.
Wind components are scaled by the maximal wind speed. Kelvin wave is the slowest eastwardpropagating eigensolution of the linearized primitive equations on the sphere, associated with a large part of atmosphere and ocean variability in the tropics. Together with the MRG wave, Kelvin wave fills the frequency gap between the Rossby and inertiagravity modes in the tropics. In the case of linearized equations on the equatorial betaplane, the Kelvin wave is sometimes denoted n=1 eastwardpropagating inertiagravity mode.
(b) Mixed Rossbygravity Waves
Horizontal structure of the n=0 westwardpropagating rotational mode on the sphere, mixed Rossbygravity wave.
Geopotential height is scaled by the maximal height. Colours vary between 1 and +1 with positive/negative perturbations in red/blue.
Wind components are scaled by the maximal wind speed. Mixed Rossbygravity (MRG) wave is the equatoriallytrapped mode which together with the Kelvin wave fills the frequency gap between the Rossby and gravity modes in the tropics. As its zonal wavenumber increases, the MRG wave becomes increasingly more rotational mode. In the case of linearized equations on the equatorial betaplane, the mixed Rossbygravity wave is denoted the westwardpropagating MRG mode.
(c) Eastwardpropagating Inertiagravity Waves
Horizontal structure of the slowest eastwardpropagating inertiagravity (EIG) mode on the sphere, n=0 EIG wave. Geopotential height is scaled by the maximal height. Colours vary between 1 and +1 with positive/negative perturbations in red/blue. Wind components are scaled by the maximal wind speed.
(d) Rossby Waves
Horizontal structure of the fastest Rossby mode on the sphere, n=1 Rossby wave.
Geopotential height is scaled by the maximal height. Colours vary between 1 and +1 with positive/negative perturbations in red/blue.
Wind components are scaled by the maximal wind speed. The n=1 Rossby wave, together with the Kelvin wave, provides basic understanding of largescale circulation in response to tropical heating perturbations, following Gill (1980) and the longwave approximation.
(e) Westwardpropagating Inertiagravity Waves
Horizontal structure of the slowest westwardpropagating inertiagravity (WIG) mode on the sphere, n=0 WIG wave. Geopotential height is scaled by the maximal height. Colours vary between 1 and +1 with positive/negative perturbations in red/blue. Wind components are scaled by the maximal wind speed.
The Effect of the Background Flow on the Meridional Structure of the Hough Functions
Meridional structures of the Hough functions for the zonal wavenumber k=1 and equivalent depth 10 km. The comparison is between the case of the background state at rest and the background flow derived from ERA5 reanalysis data for level 500 hPa and MAM season.
These results reproduce ones from Kasahara, A., 1981: Corrigendum, J. Atmos. Sci. 38, 22842285. The original article is Kasahara, A., 1980: Effect of Zonal Flows on the Free Oscillations of a Barotropic Atmosphere, J. Atmos. Sci. 37, 917929.
Hough Harmonics
These and further Hough harmonics are available for two projections:
https://modes.cen.unihamburg.de/ebla/MODES/Robinson and
https://modes.cen.unihamburg.de/ebla/MODES/EquidistantCylindrical
Feel free to use them. Please acknowledge the MODES project webpage.
Kelvin Waves 


Mixed Rossbygravity Waves 

Eastwardpropagating Inertiagravity Waves 

Rossby Waves 

Westwardpropagating Inertiagravity Waves 
