On the Theory of the Diurnal Tide

Laplace's tidal equation for diurnal tides of longitudinal number one is investigated. It is found that in addition to the previously found solutions (Hough Functions) corresponding to positive equivalent depths there are also Hough Functions corresponding to negative equivalent depths. Both are necessary for the representation of observed tidal data. As an application of the Hough Functions the diurnal surface pressure oscillation resulting from diurnal variations in insolation is computed. It is found that the insolation model due to Siebert can account for only one-third of the observed pressure oscillation. i . INTRODUCTION In the study of at,mospheric tides, the diurnal tide has been relatively neglected; the semidiurnal tide has received most, of the attention. In recent years, however, with improved rocket data, it has been found that the diurnal period is dominant in mesospheric winds and temperatures (hliers [8], Beyers and Miers [2]>. As a result a renewed study of the diurnal tide seemed important. Inevitably, the study of tides requires a knowledge of Hough Functions for the period and longitudinal wave number under consideration (Siebert [lo]). For the diurnal tide following the sun (i.e., ~ = + 1 day; longitudinal wave number = + 1) the relevant Hough Functions present some interesting problems. Haurwitz [5] has recently computed three of these functions and their associated eigenvalues (i.e., equivalent depths). He found that the computation of these functions in terms of Associated Legendre Polynomials was difficult insofar as each Hough Function required a substantial number of polynomials for its accurate representation. Moreover, the amplitude of the functions he found was confined to within a region of 30' about the equator; i.e., within the critical latitudes wheref(=29 sin 6 ) equals 29 sin 30' (=%/I day). As a result it appeared that, a t best, a great number of these functions would be needed to represent globally distributed functions. Because of the confinement to the region of 30" about the equator, it was felt by this author that an investigation of diurnal tides on an equatorially centered @-plane would yield simplified approximations to the Hough Functions and their associated equivalent depths. The resulting calculation will not be described in this paper since its results are not explicitly used. However, as was expected, good approximations to all the Hough Functions of the 1 Present address National Center for Atmospheric Research, Boulder, COlo. type found by Haurwite and their associated equivalent depths were found. More interesting, however, was the fact that these were not the only eigensolutions. There was also an infinite set of eigenfunctions whose amplitude was concentrated outside the critical latitudes. The 6-plane approximation is, of course, extremely bad a t these latitudes. On the other hand, the inclusion of these eigenfunctions is necessary if the total set of functions is to be complete. This, in turn, suggests that the set of eigenfunctions used by Haurwite [5] is not complete. A new investigation of Laplace's Tidal Equation for a spherical surface was therefore undertaken. The remainder of this paper deals with that investigation, its results, and their application. It was found that in addition to the eigensolutions found by Haurwitz, there is, in fact, another infinite set of eigenfunctions whose amplitude is concentrated outside the critical latitudes. Associated with these eigensolutions are negative equivalent depths.2 The existence of these eigensolutions proves the incompleteness of the original set. Moreover, the set of Hough Functions, including the new ones, proves quite suitable for the representation of observed tidal distributions. Finally, the Hough Functions are used in order to investigate the diurnal surface pressure oscillation that should result from diurnal variations in solar insolation.