`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: cdoswell@earthlink.net `

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 **W**_{h} 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 = C_{p}lnq). 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, z_{q}, 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} m^{2} 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.

**
**

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.

- The
**first**is having to write about the interaction between synoptic scale processes and severe convection. This has forced me to learn more about PV in general and how the concepts work. - The
**second**is teaching my graduate course in "Advanced Forecasting Techniques" and having to address the topic in some reasonably balanced way. - The
**third**is my attendance at the recent Cyclone Workshop in Val Morin, Quebec, where I met and talked with a number of PV/IPV advocates.

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.