Abstract
By using the method of multiple scales in height and a variety of methods in latitude, analytic solutions for equatorial waves in combined vertical and horizontal shear are derived. In contrast to the formulation of Andrews and McIntyre (1976b), latitudinal shear is incorporated at lowest order in the vertical shear expansion, showing that it is unnecessary to carry the calculation to first explicit order in Ri−½, where Ri is the Richardson number. The multiple-scales approximation implies that, with the exception of the overall amplitude factor and arbitrary overall constant phase factor, all properties of the wave at the height z=z0 are determined by the local wind profile, V(y)=U(y,z0). In consequence, understanding waves in two-dimensional shear reduces to the much simpler problem of solving the one-dimensional eigenvalue equation in latitude which is derived by assuming that the mean wind is V(y), a function of latitude only. This is done using ordinary perturbation theory, a non-perturbative analytic procedure and the Hermite spectral method for various classes of waves. Once its solutions are known, the overall amplitude factor may be found by using the wave action equation as shown in the text. When the method of multiple scales is invalid, as appears true of the tropical ocean, it is shown that Hermite spectral methods in latitude are much more accurate (at least in the absence of coastal boundaries) than the finite-difference methods used in the past. The techniques discussed here are applied to several classes of observed atmospheric equatorial waves in Part II (Boyd, 1978a).
