A mathematical basis for Bretherton's (1966) view of baroclinic instability as the interaction of two counter-propagating Rossby waves (CRWs) is given for a general zonal flow for which potential vorticity (PV) is materially conserved. The two CRWs are constructed from a pair of growing and decaying normal modes so that the material displacements associated with each of the waves (a) have zero tilt (b) are orthogonal with respect to the density weighted Euclidean norm, and (c) are localized in regions of large PV gradients, by the requirement that each CRW corresponds to an extremum of wave action. As a consequence of the above constraints, the displacements associated with the CRWs are orthogonal with respect to wave action.
Although each CRW could not continue to exist alone, they can together describe the time development of any flow whose initial conditions can be described by the pair of normal modes. The CRWs' dynamical equations reduce to a simple set involving only two parameters, describing the mutual interaction of the waves and the difference in their propagation speeds. The phase-locking configuration, and normal mode growth, is possible only if the mean zonal wind and the PV gradient are positively correlated at the two CRWs locations. Hence, the obtained local conditions on the locations of the CRWs are equivalent to the integral Charney and Stern (1962) and Fjortoft (1951) conditions for instability. A more general initial value problem is also easily discussed in which there is the possibility of transient super-modal growth.
The theory is illustrated by applying it to Charney's (1947) model of baroclinic instability. Combining CRW theory with the fact that the strength of the wind field attributable to interior PV, increases with a higher power of wavelength than the strength of the wind field attributable to surface temperature, leads to a straightforward explanation of the long wavelength neutral modes of the Charney problem.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics