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[MIS16-P04] Eigenvalue analysis of atmospheric free oscillations under the influence of a zonal mean field
Keywords:atmospheric free oscillation, eigenvalue analysis, Lamb mode
Atmospheric free oscillations are normal modes of the Earth’s atmosphere. Sakazaki and Hamilton (2020) comprehensively detected an array of free oscillation modes including high-frequency modes by the spectral analysis using ERA5 reanalysis data. The frequencies and spatial structures of these modes are consistent with those predicted theoretically by the classical tidal theory, but some differences can be seen. This is due to the fact that the classical tidal theory assumes a stationary atmosphere as the basic field. Kasahara (1980) carried out a linear eigenvalue analysis for the zonal mean zonal wind in a barotropic atmospheric model and investigated the effect of the zonal wind on the free oscillations. However, the analysis method cannot take into account the vertical dependence of the basic field. Therefore, a linear eigenvalue analysis of a zonal mean zonal wind and temperature field is performed in the present study.
The primitive equations in the sigma coordinate system are used as the governing equations and linearized with respect to the zonal mean fields of zonal wind and temperature. Based on the three-dimensional spectral method of Ishioka, et al. (2022), the linearized equations are transformed into a matrix form by expanding each variable in the latitudinal direction using the associated Legendre function and in the sigma direction using the Legendre polynomial. This is done under the assumption of a sinusoidal solution for the longitude-time dependence, and the Galerkin method is then applied. For each zonal wavenumber, the eigenvalues and eigenvectors of this matrix are calculated to obtain the eigenfrequencies and spatial structures of the Lamb modes deformed by the zonal mean fields. In such an analysis, in addition to the Lamb modes, other eigenmodes, such as spurious modes due to the upper boundary of the numerical model, are also obtained. Therefore, Rayleigh friction and Newtonian cooling, which act strongly in the upper layers of this model, are introduced to increase the damping rate of such modes, and conditions for the vertical phase structure, surface horizontal structure, and damping rate are imposed to identify the Lamb modes. Note that the meridional field used as the basic field for the eigenvalue analysis is obtained by temporally and zonally averaging the ERA5 zonal wind and temperature data over 10 years.
The frequencies of eigenmodes obtained from the above analysis differ slightly from those calculated by the classical tidal theory. This indicates that the phase speeds of the eastward propagating modes increase and those of the westward propagating modes decrease due to the background field compared to the theoretical solution. This result is in good agreement with Sakazaki and Hamilton (2020) and can be mainly attributed to the Doppler shift caused by the mid-latitude jets in the troposphere. However, for the gravest Rossby mode of zonal wavenumber 1, the phase speed is faster than the theoretical solution although it is a westward propagating mode. This faster phase speed is considered to be due to the fact that the effect of latitudinal gradient of temperature field exceeds that of the zonal wind, as also pointed by Kasahara (1980). The spatial structures of the eigenmodes obtained in the present study are generally consistent with those obtained by the classical tidal theory. However, for low-frequency modes, equatorial asymmetry in the horizontal structure and westward tilt of the vertical phase are observed. The causes of such discrepancies with the theoretical solutions are also discussed.
References: [1] Ishioka, K., et al., 2022, J. Meteor. Soc. Japan, 100, 445-469. [2] Kasahara, A., 1980, J. Atmos. Sci., 37, 917-929. [3] Sakazaki, T., & K. Hamilton, 2020, J. Atmos. Sci., 77, 2519-2539.
The primitive equations in the sigma coordinate system are used as the governing equations and linearized with respect to the zonal mean fields of zonal wind and temperature. Based on the three-dimensional spectral method of Ishioka, et al. (2022), the linearized equations are transformed into a matrix form by expanding each variable in the latitudinal direction using the associated Legendre function and in the sigma direction using the Legendre polynomial. This is done under the assumption of a sinusoidal solution for the longitude-time dependence, and the Galerkin method is then applied. For each zonal wavenumber, the eigenvalues and eigenvectors of this matrix are calculated to obtain the eigenfrequencies and spatial structures of the Lamb modes deformed by the zonal mean fields. In such an analysis, in addition to the Lamb modes, other eigenmodes, such as spurious modes due to the upper boundary of the numerical model, are also obtained. Therefore, Rayleigh friction and Newtonian cooling, which act strongly in the upper layers of this model, are introduced to increase the damping rate of such modes, and conditions for the vertical phase structure, surface horizontal structure, and damping rate are imposed to identify the Lamb modes. Note that the meridional field used as the basic field for the eigenvalue analysis is obtained by temporally and zonally averaging the ERA5 zonal wind and temperature data over 10 years.
The frequencies of eigenmodes obtained from the above analysis differ slightly from those calculated by the classical tidal theory. This indicates that the phase speeds of the eastward propagating modes increase and those of the westward propagating modes decrease due to the background field compared to the theoretical solution. This result is in good agreement with Sakazaki and Hamilton (2020) and can be mainly attributed to the Doppler shift caused by the mid-latitude jets in the troposphere. However, for the gravest Rossby mode of zonal wavenumber 1, the phase speed is faster than the theoretical solution although it is a westward propagating mode. This faster phase speed is considered to be due to the fact that the effect of latitudinal gradient of temperature field exceeds that of the zonal wind, as also pointed by Kasahara (1980). The spatial structures of the eigenmodes obtained in the present study are generally consistent with those obtained by the classical tidal theory. However, for low-frequency modes, equatorial asymmetry in the horizontal structure and westward tilt of the vertical phase are observed. The causes of such discrepancies with the theoretical solutions are also discussed.
References: [1] Ishioka, K., et al., 2022, J. Meteor. Soc. Japan, 100, 445-469. [2] Kasahara, A., 1980, J. Atmos. Sci., 37, 917-929. [3] Sakazaki, T., & K. Hamilton, 2020, J. Atmos. Sci., 77, 2519-2539.