Garfinkel Chaim I, Fouxon Itzhak, Shamir Ofer, Paldor Nathan
Fredy and Nadine Herrmann Institute of Earth Sciences The Hebrew University of Jerusalem Israel.
Q J R Meteorol Soc. 2017 Apr;143(704):1554-1564. doi: 10.1002/qj.3025. Epub 2017 Apr 24.
Observational evidence for an equatorial non-dispersive mode propagating at the speed of gravity waves is strong, and while the structure and dispersion relation of such a mode can be accurately described by a wave theory on the equatorial β-plane, prior theories on the sphere were unable to find such a mode except for particular asymptotic limits of gravity wave phase speeds and/or certain zonal wave numbers. Here, an ad hoc solution of the linearized rotating shallow-water equations (LRSWE) on a sphere is developed, which propagates eastward with phase speed that nearly equals the speed of gravity waves at all zonal wave numbers. The physical interpretation of this mode in the context of other modes that solve the LRSWE is clarified through numerical calculations and through eigenvalue analysis of a Schrödinger eigenvalue equation that approximates the LRSWE. By comparing the meridional amplitude structure and phase speed of the ad hoc mode with those of the lowest gravity mode on a non-rotating sphere we show that at large zonal wave number the former is a rotation-modified counterpart of the latter. We also find that the dispersion relation of the ad hoc mode is identical to the n = 0 eastward propagating inertia-gravity (EIG0) wave on a rotating sphere which is also nearly non-dispersive, so this solution could be classified as both a Kelvin wave and as the EIG0 wave. This is in contrast to Cartesian coordinates where Kelvin waves are a distinct wave solution that supplements the EIG0 mode. Furthermore, the eigenvalue equation for the meridional velocity on the β-plane can be formally derived as an asymptotic limit (for small (Lamb Number)) of the corresponding second order equation on a sphere, but this expansion is invalid when the phase speed equals that of gravity waves i.e. for Kelvin waves. Various expressions found in the literature for both Kelvin waves and inertia-gravity waves and which are valid only in certain asymptotic limits (e.g. slow and fast rotation) are compared with the expressions found here for the two wave types.
有强有力的观测证据表明，赤道存在一种以重力波速度传播的非色散模式。虽然这种模式的结构和色散关系可以用赤道β平面上的波动理论精确描述，但此前球面上的理论除了重力波相速度的特定渐近极限和/或某些纬向波数外，无法找到这样的模式。在此，我们推导出球面上线性化旋转浅水方程（LRSWE）的一个特设解，它以几乎等于所有纬向波数下重力波速度的相速度向东传播。通过数值计算以及对近似LRSWE的薛定谔特征值方程进行特征值分析，在求解LRSWE的其他模式背景下，阐明了该模式的物理解释。通过将特设模式的经向振幅结构和相速度与非旋转球面上最低重力模式的进行比较，我们表明在大纬向波数时，前者是后者的旋转修正对应物。我们还发现，特设模式的色散关系与旋转球面上n = 0的向东传播惯性重力（EIG0）波相同，该波也几乎是非色散的，所以这个解既可以归类为开尔文波，也可以归类为EIG0波。这与笛卡尔坐标系不同，在笛卡尔坐标系中，开尔文波是补充EIG0模式的一种独特波动解。此外，β平面上经向速度的特征值方程可以形式上推导为球面上相应二阶方程的渐近极限（对于小（兰姆数）），但当相速度等于重力波速度时，即对于开尔文波，这种展开是无效的。将文献中仅在某些渐近极限（如慢旋转和快旋转）下有效的开尔文波和惯性重力波的各种表达式，与这里找到的这两种波型的表达式进行了比较。
