Tidal Spherical Harmonics

The convention used here for spherical harmonics of tidal elevations is similar to that used by Lambeck (1980, eq. 6.2.1--allowing for a misprint in his 6.2.1c), except for some changes in phase for diurnal tides. Like Lambeck, we use strictly phase lags in the tidal arguments. The usual convention in satellite geodesy is to use phase leads. Generally, most of the tidal literature uses unnormalized coefficients, which we also employ here.

For any given harmonic constituent of frequency $\sigma$, we express the tidal height fluctuation $\zeta$ (reckoned relative to the seabed) as

$$\begin{eqnarray*} \zeta & = & \sum_n \sum_m \left\{ \rule{0em}{1.1em} D_{nm}^{... ... \psi_{nm}^{-} \rule{0em}{1.1em} \right\} \, P_n^m(\cos\theta), \end{eqnarray*}$$

comprising prograde $(+)$ and retrograde $(-)$ waves of amplitude $D_{nm}^\pm$ and phase lag $\psi_{nm}^\pm$, where $(\theta,\phi)$ are spherical polar coordinates and $P_n^m(\mu)$ is an associated Legendre function (unnormalized). Note that in this context, ``prograde'' means in the direction of the tide-raising body, i.e., westward; this usage is standard in tidal literature but is opposite to that used in polar motion studies. The $D_{nm}^{\pm}$ and $\psi_{nm}^{\pm}$ parameters are deduced from numerical hydrodynamic models or from satellite altimeter measurements of the global tide; this is done by direct numerical quadrature over the globe (here assumed spherical) of the tidal elevations. Some of the parameters (mainly those of order 1 for diurnal tides and 2 for semidiurnal) can be deduced from analysis of satellite orbit perturbations.

The variable $t$ is time reckoned from some conventional point that is generally different for each tidal constituent. For example, for M$_{2}$ time is reckoned from the instant the mean moon passes the Greenwich meridian. More clearly, the argument $\sigma t$ is equal to a linear combination of the Brown astronomical longitudes:

$$\sigma t = \sum_{i=1}^6 \beta_i\, \ell_i(t) + \chi$$

where the $\beta_i$ are the Doodson numbers for the given tidal constituent and $\ell_i(t)$ are the astronomical longitudes:

$$\begin{eqnarray*} \ell_1: & & \tau - \mbox{mean Greenwich lunar time}\\ \ell_... ...de}\\ \ell_6: & & p' - \mbox{mean longitude of solar perigee.} \end{eqnarray*}$$

The additional angle $\chi$ is employed so that we may consistently use cosine functions above and also maintain positive amplitudes; this is standard following Doodson's convention. For the primary tides we have $\chi = 0$ for the semidiurnals M$_{2}$, N$_{2}$, S$_{2}$, K$_{2}$, (but 180$^{\circ}$ for the small tide R$_2$); $\chi = 0$ for the long-period tides Sa, Ssa, Mm, Mf, Mt; but $\chi = -90$$^{\circ}$ for Q$_{1}$, O$_{1}$, P$_{1}$, and $+90$$^{\circ}$ for K$_{1}$.

Lambeck, K., The Earth's Variable Rotation, Cambridge Univ. Press, 1980.

Doodson, A. T. and H. D. Warburg, Admiralty Manual of Tides, HMSO, 1941.