The normal modes of Laplace's tidal equations seem to be a consolidated subject in the literature. However, up until now, no mode for the zonal wavenumber zero, (i.e., independent of longitude) had been identified as a mixed Rossby–gravity mode with physical meaning. Moreover, the meridional structures of westward gravity modes were not considered to be similar to the meridional structures of the corresponding inertio‐gravity modes for non‐zero wavenumbers (k>0). Here, a mixed Rossby–gravity mode with zero zonal wavenumber () is identified for the first time, and a new classification of the normal modes for the zonal wavenumber zero is proposed. Under the new classification, all the meridional structures corresponding to zero wavenumber modes are similar to the meridional structures of their respective non‐zero wavenumber modes. Moreover, using the new classification, all modes of a given type show smooth frequency variations with the zonal wavenumber, which approach asymptotically the analytic dispersion curves from the ‐plane shallow‐water equations as the fluid height tends to zero.

中文翻译：

---

关于纬向波数为零的拉普拉斯潮汐方程的简正模态指数

拉普拉斯潮汐方程的正规模态似乎是文献中的一个综合主题。然而，到目前为止，尚未将零纬向波数模式（即与经度无关）确定为具有物理意义的混合罗斯贝重力模式。此外，对于非零波数（k>0），西向重力模式的子午结构不被认为与相应惯性重力模式的子午结构相似。在这里，首次识别了带状波数为零的罗斯贝-重力混合模式（），并提出了带状波数为零的简正模式的新分类。在新的分类下，所有零波数模式对应的子午结构都与其各自的非零波数模式的子午结构相似。此外，使用新的分类，给定类型的所有模式都显示出随纬向波数的平滑频率变化，当流体高度趋于零时，渐近地接近平面浅水方程的解析色散曲线。