Updated: 23 September 2006 - moved over from its former site and revised slightly.
There's been a great deal of interest expressed recently in the meteorological literature about Isentropic Potential Vorticity (or IPV). This discussion is aimed at expressing myself on the subject and at giving anyone interested some feel for the subject, hopefully without overwhelming them. As always, comments are welcome - send them to: email@example.com
Most everyone involved with weather and weather forecasting knows something about vorticity. When vorticity is discussed, it is usually the vertical component, z, of the vector vorticity, W, that is under consideration.
where Wh is the horizontal vorticity vector and k is the unit vector in the vertical. As many folks know, the equation that describes the change of z following a parcel (i.e., the Lagrangian form of the so-called vorticity equation) is, in pressure coordinates,
where w is the vertical motion in p-coordinates, f is the Coriolis parameter, and F is the frictional force. From (2) it should be clear that vorticity is not conserved following a parcel, in general. Only under conditions where the right-hand side of (2) vanishes would vorticity be conserved.
As discussed in Holton (1992, p. 98 ff.), the term potential vorticity is attached to a number of concepts with slightly different shadings of meaning, but they all refer to the ratio of the vorticity to some measure of vertical depth in the atmosphere. The ultimate generalization of this concept is the so-called Ertel's potential vorticity, Z, defined by
where r is density and s is the entropy (related to the potential temperature, q, by s = Cplnq). Note that the vorticity and the gradient operator in (3) are both three-dimensional. Under the condition of frictionless, adiabatic flow, it can be shown that Z is conserved following a parcel; that is,
when F = 0 and Dq/Dt=0.
Isentropic Potential Vorticity
In isentropic coordinates, potential vorticity is expressed as
The vorticity, zq, in this expression is evaluated on an isentropic surface; g is the acceleration due to gravity. It can be shown that (5) is simply a special case of the more generalized form (3), and for adiabatic, frictionless flow, DP/Dt=0. That is, P is conserved. Consider the units of P:
where the factor of 10-2 comes in because of the conversion of pressures in millibars (or hectopascals, if you insist) to MKS units. As noted in Bluestein (1993, p. 182), a value for P of 10-6 °K kg-1 m2 s-1 is called one IPV unit (IPVU=1.0). P-values of around one IPV unit are associated with the tropopause, with higher values above (due to the great static stability of the stratosphere), and lower values below.
Clearly, it is of considerable value to have a conserved quantity in understanding the dynamics. It turns out that, subject to some assumptions about the flow and a knowledge of some boundary conditions, one need only have the P-field in three dimensions to determine both the wind field and the stability ( ) field; this is the invertibility principle associated with IPV and is a powerful tool in some dynamical contexts.
Is it worth the trouble to learn IPV thinking?
What follows is my not-necessarily unbiased opinion about the use of IPV as a way to understand the atmosphere. Although there have been some papers published advocating the use of IPV as a way to think about the atmosphere (notably, Hoskins et al. 1985). Bluestein (1993, p. 202) offers an interesting commentary on IPV thinking, with which I agree:
... This approach is neither more nor less accurate than the quasigeostrophic diagnosis in pressure coordinate: "IPV thinking" merely provides another perspective. (If so, why do we even bother to discuss it? We do so because at the time of this writing it is in vogue! If new insights into the behavior of synoptic-scale systems are found using "IPV thinking," then it will become very important. On the other hand, if no new insights are found, then it may fade away ... .
IPV thinking probably will not "fade away" as long as it has its advocates, but it might well see limited use if it does not offer new insights. Like Bluestein, I know of no particularly compelling new notions rooted in IPV thinking; rather, what one sees is an alternative perspective and one that requires a lot of thought and hard work to develop. I cannot recommend that a forecaster make the effort, especially since the basic physics of synoptic weather systems is relatively easy to grasp using the classical quasigeostrophic dynamics concepts already presented elsewhere in this material. Nevertheless, it is potentially useful to stay up with what is current in the field, and I certainly cannot preclude the development of useful new concepts rooted in IPV thinking.
Update #1: (added on 07 August 1997).
Although my opinion about how important it is for an operational forecaster to learn IPV thinking has not changed, I am now beginning to see some uses for it in the context of developing an understanding of cyclones and their associated vertical motions. The IPV inversion process allows the visualization of the flows (including the vertical motions) of vortices, showing their "reach" both horizontally and vertically. In principle, this is nothing more than what can be obtained from previous methods, although it might be rather more cumbersome to do so. The inversion process is by no means trivial, involving the solution of partial differential equations (and the specification of boundary conditions to allow such a solution). Hence, it is unlikely to become a useful real-time forecasting tool, although it might well prove useful in research and post-mortem case studies.
Three things have conspired to continue an evolution in my viewpoint on this topic.
I have not changed my ideas about IPV in the sense described in Howie's book. As it is applied, it really is not a whole lot different from QG theory and the whole conventional, isobaric point of view. Moreover, if you actually compare vorticity and potential vorticity maps on isobaric surfaces, the patterns typically are almost indistinguishable. The spatial variation of static stability on an isobaric surface is usually small enough that its incorporation into the isobaric PV is not going to alter the pattern much from the conventional vorticity, except perhaps when the tropopause cuts through the isobaric surface.
However, I am learning to appreciate the value of isentropic analysis, in general. [I've known for a long time that isentropic surfaces are ideal for following moisture, since adiabatic processes mean that a parcel must stay on an isentropic surface ... a condition which is absolutely not true for an isobaric surface!] Analyses on isentropic surfaces can be very different from analyses on isobaric surfaces, the latter of which are quasi-horizontal. In some situations, the isentropic surfaces can be very tilted away from the horizontal ... when the lapse rates are at or near the dry adiabatic value, the isentropic surfaces are very nearly vertical!
For some purposes, IPV analysis has considerable value. In particular, for following disturbances in the vicinity of the tropopause, IPV is clearly a tool of choice. I also like the idea of dynamic tropopause maps, where the dynamic tropopause (or, "DT") is defined to be a surface of IPV equal to some value, such as 1.0 or 1.5 IPVU, and the field analysed is the temperature or pressure associated with that value. When the tropopause is steeply inclined, implying an intrusion of stratospheric air into the troposphere, this means a strong gradient in, say the pressure of the DT. We typically find upper tropospheric jet streams along this gradient.
The importance of following IPV anomalies near the tropopause becomes clear when we consider that real cyclogenesis is dominated by the interaction between upper level and lower level PV anomalies. The "penetration depth" of an IPV anomaly is governed strongly by the static stability distribution ... a strong anomaly at upper levels passing over a region of high static stability does not induce much of a disturbance beneath that layer of strong static stability. However, the anomaly circulations also induce advections that alter the wind and thermal fields, so IPV anomalies are just not something passively being carried along in the flow. Rather, they modify that flow, and the modifications, in turn, induce further changes. This complex, nonlinear process is best viewed in the IPV framework, I am coming to believe.
Closed centers of IPV (or closed centers of the pressure at the DT) have associated with them a set of winds and a static stability field that can be determined by the PV inversion technique, that couples the observed IPV distribution with a "balance condition" (e.g., the geostrophic wind law, or the gradient wind law, or a number of other candidate balance conditions) and a set of boundary conditions to define the wind and static stability fields. Thus, the flows associated with a particular IPV anomaly can be determined ... as noted earlier, this makes possible an experiment where it can be seen how the atmosphere would have evolved in the absence of this IPV anomaly. Being able to test such hypotheses is a good thing, and using IPV makes this possible in a relatively simple and elegant way ... although I see the actual IPV inversion to be a non-trivial process. In spite of the rather complicated nature of IPV inversion, it is quite a bit more difficult to do this sort of "anomaly removal" in any of the more conventional frameworks.
From a tutorial perspective, many of us find the "conventional" framework most compelling to our "intuition" ... however, it is not really "intuition" that is involved here! It's habits, and what we grew up with in our meteorological education. I suspect we might find IPV just as intuitive and compelling if that had been what we learned right from the outset. Old habits die hard, and old perspectives can linger well beyond their period of usefulness. IPV may just be a fad, but I am now less inclined to believe that.
Furthermore, if I were wanting to do estimation of the QG (or other balance conditions) processes associated with vertical motion, it is possible to make dynamically correct inferences of vertical motion from the advection of IPV. These inferences include both the conventional vorticity advection and thermal advection contributions associated with the QG Omega Equation. In this way, IPV approaches are yet a third alternative way to do such estimation, since Q-vectors are a second way to combine the separate terms in the QG Omega Equation. Since solution of the Q-vector form of the Omega Equation still requires inverting an elliptic differential equation, with boundary conditions, I think IPV advection is arguably the simplest and most elegant way to develop a qualitative assessment of QG-forced vertical motion.
I may have more updates to this page in the future. Stay tuned.
Bluestein, H.R., 1993: Synoptic-Dynamic Meteorology in Midlatitudes. Vol. II: Observations and Theory of Weather Systems . Oxford University Press, 594 pp.
Hakim, G.J., D.Keyser, and L.F. 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.
Holton, J.R., 1992: An Introduction to Dynamic Meteorology (3rd Ed.), Academic Press, 511 pp.
Hoskins, B.J., M.E. McIntyre, and A.W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc ., 111, 877-946.
Raymond, D.J., 1992: Nonlinear balance, and potential-vorticity thinking at large Rossby number. Quart. J. Roy. Meteor. Soc., 118, 987-1015.