Approximations to the scattering of water waves by steep topography

A new method is developed for approximating the scattering of linear surface gravity waves on water of varying quiescent depth in two dimensions. A conformal mapping of the fluid domain onto a uniform rectangular strip transforms steep and discontinuous bed profiles into relatively slowly varying, smooth functions in the transformed free-surface condition. By analogy with the mild-slope approach used extensively in unmapped domains, an approximate solution of the transformed problem is sought in the form of a modulated propagating wave which is determined by solving a second-order ordinary differential equation. This can be achieved numerically, but an analytic solution in the form of a rapidly convergent infinite series is also derived and provides simple explicit formulae for the scattered wave amplitudes. Small-amplitude and slow variations in the bedform that are excluded from the mapping procedure are incorporated in the approximation by a straightforward extension of the theory. The error incurred in using the method is established by means of a rigorous numerical investigation and it is found that remarkably accurate estimates of the scattered wave amplitudes are given for a wide range of bedforms and frequencies.

Equatorial waves in the lower stratosphere. II: Annual and interannual variability

The annual and interannual variability of idealized, linear, equatorial waves in the lower stratosphere is investigated using the temperature and velocity fields from the ECMWF 15-year re-analysis dataset. Peak Kelvin wave activity occurs during solstice seasons at 100 hPa, during December-February at 70 hPa and in the easterly to westerly quasi-biennial oscillation (QBO) phase transition at 50 hPa. Peak Rossby-gravity wave activity occurs during equinox seasons at 100 hPa, during June-August/September-November at 70 hPa and in the westerly to easterly QBO phase transition at 50 hPa. Although neglect of wind shear means that the results for inertio-gravity waves are likely to be less accurate, they are still qualitatively reasonable and an annual cycle is observed in these waves at 100 hPa and 70 hPa. Inertio-gravity waves with n = 1 are correlated with the QBO at 50 hPa, but the eastward inertio-gravity n = 0 wave is not, due to its very fast vertical group velocity in all background winds. The relative importance of different wave types in driving the QBO at 50 hPa is also discussed. The strongest acceleration appears to be provided by the Kelvin wave while the acceleration provided by the Rossby-gravity wave is negligible. Of the higher-frequency waves, the westward inertio-gravity n = 1 wave appears able to contribute more to the acceleration of the 50 hPa mean zonal wind than the eastward inertio-gravity n = 1 wave.

Equatorial waves in the lower stratosphere. I: A novel detection method

Wavenumber-frequency spectral analysis and linear wave theory are combined in a novel method to quantitatively estimate equatorial wave activity in the tropical lower stratosphere. The method requires temperature and velocity observations that are regularly spaced in latitude, longitude and time; it is therefore applied to the ECMWF 15-year re-analysis dataset (ERA-15). Signals consistent with idealized Kelvin and Rossby-gravity waves are found at wavenumbers and frequencies in agreement with previous studies. When averaged over 1981-93, the Kelvin wave explains approximately 1 K-2 of temperature variance on the equator at 100 hPa, while the Rossby-gravity wave explains approximately 1 m(2)s(-2) of meridional wind variance. Some inertio-gravity wave and equatorial Rossby wave signals are also found; however the resolution of ERA-15 is not sufficient for the method to provide an accurate climatology of waves with high meridional structure.