Piecewise potential vorticity inversion (PPVI), as used in many case studies, is a method to analyze atmospheric observations to better understand interactions of various features in the potential vorticity (PV) distribution. To the best of my knowledge, there has been as yet no discussion of the limitations and utility of this method. To initiate such a discourse, I applied (in Egger 2008) the technique of PPVI to rather simple idealized cases that are understood perfectly well. One does not expect to find new results by applying PPVI to linear one-dimensional Rossby waves or to the motion of PV anomalies in the Eady model, but PPVI ought to recover basic insights with respect to these phenomena. However, as shown in Egger 2008, PPVI fails to deliver useful results. This casts doubts on the application of PPVI to more complicated situations.
I am pleased that Methven and de Vries (2008) use this opportunity to participate in the discussion of PPVI and to contribute to a clarification of some issues. It is their main point to give up the standard technique of PPVI, which demands that various pieces of the PV “occupy distinct spatial domains.” Instead, Methven and de Vries (2008) propose “to partition PV structures that in general have spatial overlap and evolve as described by simple ordinary differential equations.”
Methven and de Vries (2008) illustrate this idea by applying the concepts of counterpropagating Rossby waves (CRWs) and kernel Rossby waves (KRWs) to various dynamical problems related to the Eady model discussed in Egger 2008. I agree that a fairly good understanding of many of these idealized flow problems can be achieved on the basis of these methods. As a matter of fact, I analyzed barotropic instability in a low-order model (Egger 2008) and found that CRWs are just as helpful in explaining this instability as are other mathematical structures. There is, however, the obvious question also posed by Methven and de Vries about what to do with these concepts in the face of an observed flow development. For example, Hakim et al. (1996) analyzed the interaction of two cyclonic flow features on the basis of PPVI with spatially fixed partitioning. It is hard to see how they could have profited from defining CRWs or KRWs. One would need a very large number of baroclinic Rossby waves to project the observed flow state on them. Any understanding of the superposition of these modes would be fairly difficult even though Hakim et al. (1996) present useful tendency calculations on the basis of standard PPVI. To take another example, I have been extending the work in Egger 2008 by looking at the two-dimensional interaction of fairly distinct PV extrema on an f plane. It turns out that PPVI gives fairly useful results in terms of streamfunction tendency for the case of one cyclonic and one anticyclonic “point vortex” but tends to fail if rather modest modifications are imposed. There are simply no CRWs on this f plane, nor would it make sense to explain this type of vortex interaction in terms of other wave modes.
It seems to me that we are just beginning to understand under which circumstances PPVI in its standard form will yield useful results. The examples in Egger 2008 demonstrate clearly that there are many cases where PPVI is misleading, but the tendency calculations of Hakim et al. (1996), for example, show in contrast that PPVI can be useful.
REFERENCES
Egger, J., 2007: Counterpropagating Rossby waves and barotropic instability. Meteor. Z., 16 , 581–585.
Egger, J., 2008: Piecewise potential vorticity inversion: Elementary tests. J. Atmos. Sci., 65 , 2015–2024.
Hakim, G., D. Keyser, and L. Bosart, 1996: The Ohio Valley wave-merger cyclogenesis event of 25–26 January 1978. Part II: Diagnosis using quasigeostrophic potential vorticity inversion. Mon. Wea. Rev., 124 , 2176–2205.
Methven, J., and H. de Vries, 2008: Comments on “Piecewise potential vorticity inversion: Elementary tests”. J. Atmos. Sci., 65 , 3003–3008.
