Potential Vorticity Equations

Potential Vorticity (PV) is a fantastic variable to examine the atmosphere first and foremost because it nicely summarizes the mass and wind fields into one single variable. It's associated governing equation of sources and sinks (i.e. the PV budget) is therefore useful in diagnosing physical processes acting to grow or decay an atmospheric disturbance. However there are some major downfalls with the typical formulations of PV budgets.

The much celebrated Haynes and Mcintyre (1987) [HM87] presented a PV substance equation on isentropic coordinates (in flux form):

where Q is isentropic PV, F is frictional torque and all other variables have their typical meteorological meanings. This equation is rightfully celebrated because it essentially states that PV is conserved following adiabatic and frictionless flow. This fact makes isentropic PV extremely useful since diabatic and adiabatic components can be easily separated and such processes diagnosed in the dynamics of phenomena of interest.

However, the first stumbling block for me with this equation has unfortunately always been the isentropic nature of the equation itself. Isentropes are extremely limiting when studying, arguably, most phenomena of interest (these days). Why so? Because they aren't as well behaved as surfaces like pressure or height. In many instances isentropes stray very far from the horizontal and even become folded. When this happens close to the ground, they often intersect the surface, and PV is no longer conserved. This is extremely limiting in the case of any system that has strong temperature gradients and/or is close to the surface, which encompasses, arguably, most or at least many phenomena of interest. For me, the issue has been dealing with the Saharan boundary layer which is so well mixed, isentropes are vertical up to 700hPa, but then at night becomes more strongly stratified. This diurnal fluctuation in isentropes also makes them extremely hard to work with since there is no consistent isentrope that represents a certain level when examining phenomena on synoptic (or even mesoscale) time scales.

When it comes to PV, I often find that this issue of using isentropic surfaces is often glossed over. Tory et al. (2012) [T2012]explicitly recognizes this issue and presents a set of equations in geometric coordinates so as to be universally useful. This somewhat negates the extremely useful property of conservation of PV following isentropic flow, but I would argue that in many situations, that property has already been negated by the above.

While these issues with isentropes are often glossed over, it's clearly recognized since many studies use PV budget equations that are not of the HM87 form, such as in T2012:

in geometric coordinates, or a 3-dimensional version of the isobaric PV budget in Zhang and Ling (2012):

I've used the latter a lot since data is often most readily available in isobaric coordinates. However, as discussed by T2012 extensively, these forms run into one major issue; the separation of diabatic and adiabatic sources and sinks is no longer perfect. T2012 and others typically refer to this as adiabatic/diabatic cancellation. The issue is most prevalent when comparing the vertical advection of PV and the diabatic source term. This process is depicted by T2012:

By adding heat to the mid-levels of the atmosphere the mass forcing (MF; diabatic term) sinks the mid-level isentrope, thereby increasing the static stability below, decreasing it above, and thus increasing PV below, and decreasing it above. This is depicted in b. However, since diabatic heating is nearly always associated with vertical motion, the vertical advection of PV cannot be ignored. Since PV encompasses the mass and wind fields, this vertical advection of PV (a part of the CAE term in T2012) redistributes mass into the upper-level from the lower-level. This is depicted in c. Thus, there is a cancellation between the diabatic and vertical advection terms because we have vertical advection of mass across the isentrope. This poses issues in the use of the latter two equations since the diabatic forcing is no longer cleanly separated from the adiabatic forcing.

T2012 attempts to remedy this issue by separating these into terms that can be attributed directly to adiabatic or diabatic processes:

This equation provides two terms that represent a direct mass change as a result of 1) mass convergence and 2) diabatic processes, and a term that represents a forcing of the wind field and any resulting response of the atmosphere to that adjustment in the wind field (what they call the wind adjustment and evolution; WAE). The WAE term can be expanded in the usual way:

or in flux form:

The first represents familiar terms that correspond to the advection, convergence, and tilting of absolute vorticity. The second represents advective and non-advective fluxes of vorticity. Since they are all dotted with the gradient of potential temperature these all represent processes perpendicular to the isentropes. Thus this essentially reorganized the PV budget equation from the traditional components (mixed mass and vorticity advection, diabatic, and frictional sources of PV) to a more specific mix of vorticity advection, convergence, and tilting relative to isentropes, mass convergence, local diabatic, and frictional sources of PV.

I still have some issues with this formulation. First mass convergence is adiabatic and when compared with the diabatic source term, this is likely to exhibit similar adiabatic/diabatic cancellation that existed when we had vertical advection of PV. However, this formulation has improved the situation separating the vorticity advection that was previously also mixed in with that term.

Overall, the PV budget formulation in T2012 provides a good means through which to separate diabatic and adiabatic sources and sinks of PV in a non-isentropic framework and is more exact in separating the sources and sinks of PV from changes in the mass and wind fields. There wasn't much in the way of literature on these issues prior to T2012 so this paper provides a great source of information and useful equations for anyone attempting to use PV in their research. Care should still be taken to select the appropriate PV formulation for the situation though.