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A study of the fluctuations of the atmospheric planetary flow patterns represented by spherical harmonics


Erik Eliasen ,

University of Copenhagen, DK
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Bennert Machenhauer

University of Copenhagen, DK
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A procedure used for expanding the height of a pressure surface over the Northern Hemisphere in a series of spherical-harmonic components is described. The corresponding spherical-harmonic representation of the stream function is obtained by utilizing the geostrophic balance condition. From this representation mean values for the spectral distribution of the kinetic energy at the 500 mb level for January 1957 is presented.

The behaviour of the components with the largest horizontal scales is considered at the 500 mb and the 1000 mb levels during the 90 days period from 1 December 1956 to 28 February 1957. In general each of these components exhibits smaller or larger fluctuations, and it is attempted to investigate the character of the shorter periodic fluctuations by eliminating the constant and the long periodic parts of the stream field. For the components with wavenumber 1, 2, and 3, and with the most large-scale meridional variation, the 24 hours tendency fields show a more or less regular westward propagation with mean values for the velocity of propagation, corresponding to periods about 5 days. For the components with the same wavenumbers but with the second largest meridional scales we find for the daily deviations from the 15 days mean flow displacements also mainly towards the west and also with mean values for the velocity of propagation, decreasing with decreasing horizontal scale. The mean values of the velocity of propagation obtained in this way for the different components are nearly in accordance with the velocities determined by the Rossby effect, especially if this effect is reduced somewhat by a weak divergence effect.

On the basis of the spectral form of the barotropic vorticity equation the time derivatives for the expansion coefficients as well as for the amplitude and phase angle of the stream function at the 500 mb level have been computed for each day in January 1957, for some components of the zonal flow and some of the components with wave-number 1, 2, 5, and 6. From these time derivatives 48 hours tendencies have been evaluated and compared with the corresponding observed ones. In general it is found that the agreement is better for components with moderately large scales than for components with very large scales. To some extent this may be explained by the neglect of quasi-stationary effects, and as a very simple attempt, these are represented by a constant term. Finally the contributions from the barotropic interactions between different groups of components to the change of kinetic energy of individual components are considered.

How to Cite: Eliasen, E. and Machenhauer, B., 2012. A study of the fluctuations of the atmospheric planetary flow patterns represented by spherical harmonics. Tellus A: Dynamic Meteorology and Oceanography, 17(2), pp.220–238. DOI:
Published on 01 Jan 2012.
Peer Reviewed


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