Blocking-Like Solutions of the Potential Vorticity Equation: Their Stability at Equilibrium and Growth at Resonance

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Abstract
Three-dimensional flows for which q=−λ(p)ψ where q is the potential vorticity, ψ the stream function and λ some arbitrary function of pressure, are examined. It is found that flows which satisfy this condition and are quite similar to atmospheric blocking patterns can be generated by the superposition of a zonal current independent of the meridional coordinate plus two eddy components. These flows, for which the Jacobian of ψ and q is zero, are of interest because 1) in the absence of forcing they constitute steady state solutions of the potential vorticity equation; and 2) the possibility exists that they can be forced resonantly to a finite amplitude by means of a potential vorticity source. The arbitrariness in the choice of λ is removed by specifying the vertical profile of the diabatic heating. It is shown that when the latter is a linear function of Pressure the resultant forced flow is nearly equivalent barotropic, stable to small amplitude perturbations, with a tendency for the blocking patterns to become somewhat move prominent with increasing pressure, in rather good agreement with observations of blocking highs. By integration of a three-level beta-plant model in time, it is shown that it is indeed possible, in the absence of dissipation, to thermally fore the above types of flows at resonance and to generate flow patterns that are quite similar to atmospheric blocking patterns. It is also shown that even when a rather broad spectrum of modes is thermally forced, the above resonant modes tend to dominate the flow, in spite of the possible interaction among modes. This would imply that provided the mean zonal flow has the proper strength to produce a resonance condition, the thermal forcing field need not have a very special structure to produce a finite amplitude disturbance through resonance.

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