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Richard S. Lindzen and Siu-shung Hong


A numerical program is developed to study the behavior of atmospheric tides in atmospheres with arbitrary distributions (with respect to altitude and latitude) of mean temperature and zonal wind. This program is used to calculate solar and lunar semidiurnal tides for various realistic models of seasonal distributions of wind and temperature.

We find that the main effect of winds on the solar semidiurnal tide (for which we have thermal excitation distributed from the ground to about 80 km) is to give rise to significant mode coupling between the main semidiurnal mode and higher modes—leading to an enhancement of the latter. The main consequences of this coupling are to (i) shift the height at which a 180° phase shift for pressure and wind fields is predicted from about 28 km (the value without winds) to substantially greater heights during the summer at middle and high latitudes; (ii) cause higher order modes to dominate semidiurnal wind oscillations near 100 km at middle and high latitudes; and (iii) reduce the amplitude at 100 km of the main semidiurnal mode by about 40% compared with what one would obtain in the absence of mean winds. All the above are shown to be consistent with recent observations.

For the lunar semidiurnal tide, the present calculations also offer new insight. If one ignores mean winds then gravitational forcing appears mathematically as a coherent drive at the earth's surface. For such forcing, previous calculations have demonstrated an immense sensitivity of tidal magnitudes to the details of the assumed vertical distribution of mean temperature. This sensitivity arose because multiple reflections due to thermal inhomogeneities could (since in the absence of horizontal temperature gradients, surfaces of constant “index of refraction” are horizontal) lead to strong constructive or destructive interference depending on small variations in temperature. The inclusion of realistic damping mitigated the sensitivity somewhat, but the sensitivity remained much greater than what one would infer from data. When one includes mean winds and horizontal temperature gradients, surfaces of constant index of refraction get “bent” and the possibilities of consistent interference diminish drastically. Indeed, our calculations show very little sensitivity to details of assumed temperature. In all cases amplitudes close to those observed are predicted

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Siu-Shung Hong and Richard S. Lindzen


A three-dimensional model is developed to study the behavior of the solar semidiurnal tide in the thermosphere. In this model, we include viscosity, thermal conductivity, Coriolis effects, the sphericity of the earth, and ion drag. Sources of excitation are absorption of solar radiation by H2O and O2 below the mesopause, and by O2 in the Schumann-Runge continuum, and O, O2, N2 in the extreme ultraviolet, in the thermosphere. The effects of mean wind below the thermosphere are obtained by joining our present results to the relevant solutions obtained by Lindzen and Hong (1974) at 100 km, modified, however, by the use of improved and corrected heating functions.

Our calculations provide detailed predictions for all meteorological fields as functions of season, solar cycle, and other parameters. Among our findings are the following:

In the present calculations, the semidiurnal tide between 100 and 130 km is dominated by the 2,4 mode excited below the thermosphere. Since the 2,4 mode decays more rapidly than the 2,2 mode above 120 km, the 2,2 mode emerges as the dominant mode above 130–200 km. During sunspot minimum conditions, the thermospheric tidal fields are driven by forcing from below, but at sunspot maximum the upward propagating tides are so severely attenuated in the lower thermosphere that thermospheric in situ forcing is of comparable importance. The thermospheric tidal fields are larger at sunspot minimum than at sunspot maximum.

Ion drag is more effective in dissipating the semidiurnal tidal modes than concluded by Lindzen (1971). The omission of either ion drag or viscosity in the model leads to unrealistic results.

Comparisons are made between our results and tidal observations at 45° latitude. In general, observed upper thermospheric amplitudes and phases are compatible with calculations. However, such comparisons are difficult below 200 km because of the importance of the 2,4 and 2,3 modes which are dependent on variable mean winds below 100 km. Indeed, we have found choices of mean wind which even lead to significant amplitudes for the 2,5 and higher order modes.

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Pi-Huan Wang, Adarsh Deepak, and Siu-Shung Hong


Formulas that can be used to determine the optical path between two points along an atmospheric ray path are derived for the case when the local zenith angle of the ray path is larger than 70°. For angles less than 70°, these formulas reduce to the airmass function; viz., the secant of the zenith angle. The formulation presented in this paper is genera] enough to be applicable to a wide variety of atmospheric conditions, such as spherical and nonspherical atmospheres, and vertically and horizontally homogeneous as well as inhomogeneous atmospheres. Formulation for the case when atmospheric refraction is important also is presented here.

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Pi-Huan Wang, Siu-Shung Hong, Mao-Fou Wu, and Adarsh Deepak


The temporal and spatial variations of the zonally-averaged ozone beating rate in the middle atmosphere on a global scale are investigated in detail based on a model study. This study shows that the mean ozone heating rate calculation can be made in a realistic manner by taking advantage of the existing two-dimensional ozone distribution and including the effect of the sphericity of the earth's atmosphere. The obtained ozone heating rates have also been Fourier-analyzed. The common features of the first three harmonic components which correspond respectively to the annual, semiannual and terannual variations are (1) the local maximum amplitudes are located in the altitude regions between 45 and 57 km; (2) local maximum amplitude can be found in the polar region; and (3) maximum horizontal gradients of the beating rate are concentrated in the high latitudes from 60 to 90°. It is also found that the amplitude of the second Fourier component at the pole is about six times greater than that at the equator.

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