such that (' = V2^. The perturbation form of Equation (5.56) is then

This is a wave equation and assuming a wave solution of the form ^ = Ael(Tt+kx+iy and substituting this into Equation (5.57) we may then derive the dispersion relation t p c = — T = u — ~r2-U. (5.58)

This expression shows that wave motion is indeed possible, and that the waves move westward relative to the zonal flow. Clearly if the zonal flow is eastward with a speed equal to this phase speed, then the Rossby wave appears stationary. The analysis may be extended to the vertical dimension also, but this is again beyond the scope of this book. For Rossby waves it can be seen that the meridional velocity variations are non-zero.

A periodic combination of westward-moving Rossby waves, and eastward-moving gravity waves, including equatorially trapped Kelvin waves is believed to be responsible for the quasi-biennial oscillation (QBO) observed in the Earth's stratosphere at equatorial latitudes. Here, mean zonal winds (between 15 mbar and 200 mbar) are found to vary quasi-periodically between eastward and westward flow with a period of about 28 months. In periods of eastward flow near the bottom of the stratosphere, equatorially trapped mixed Rossby gravity waves may propagate vertically upwards and dump their energy into an upper westward flow. As a result of this energy dumping, the region of westward flow gradually moves down. When the westward flow region reaches the bottom of the stratosphere, only Kelvin waves may propagate vertically upwards. These establish an eastward flow in the upper stratosphere where they break, which itself slowly moves down over time as more and more energy is dumped there. Eventually the eastward flow reaches the bottom of the stratosphere and mixed Rossby gravity waves begin to propagate vertically again and thus the cycle repeats itself. A similar process may be responsible for the quasi-quadrennial oscillation (QQO) observed in the equatorial stratosphere of Jupiter (discussed further in Section 5.5.3) and/or a similar semi-annual oscillation seen in Saturn's equatorial stratosphere (discussed in Section 5.6.1).

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