Potential Vorticity and the PV Perspective

Since its introduction by (Rossby, 1940) and, in its full hydrodynamic form, by (Ertel, 1942), potential vorticity (PV) has figured to a greater or lesser extent in discussions of the dynamics of the atmosphere and ocean. Today, the PV perspective is a major weapon in the armoury of atmospheric scientists when viewing data from the real atmosphere or from computer models simulating it. Hoskins et al. (1985, hereafter HMR) gave a very full account of PV, and its history and theoretical importance. The intent in this review is not to repeat this material, but rather to highlight some theoretical aspects (without detailed mathematical derivations) and some of the insights the PV perspective has given us.

The nature of the material conservation of PV can be understood through consideration of conserved quantities for an infinitesimally small material cylinder between two neighbouring isentropic (equal potential temperature, θ) surfaces and normal to them (Fig. 1). It is assumed that there are no frictional or diabatic sources. This means that the circuits given by the intersection of the cylinders with the isentropic surface remain on those surfaces and the circulation around them is conserved (Kelvin's circulation theorem). Stokes theorem then gives the absolute circulation: \begin{equation} {\rm now}\ C=\varsigma_{n}\delta S\ {\rm is materially conserved.} \end{equation} Here, \(\varsigma_n\) is the component of vorticity normal to the isentropicsurfaces.

Figure 1.  Schematic of the essential ingredients making up potential vorticity.

Mass conservation for the cylinder gives: \begin{equation} m=\rho\cdot\delta S\cdot\delta h\ {\rm is materially conserved.} \end{equation} Dividing Eq. (1) by Eq. (2) then gives: \begin{equation} \dfrac{C}{m}=\dfrac{\varsigma_{n}}{\rho\delta h}\ {\rm is materially conserved.} \end{equation}

This important result is the essence of PV. In words, it says that the spin in the normal direction divided by the separation of isentropic surfaces multiplied by density (or equivalently divided by the mass per unit area of the bounding surfaces) is conserved. To derive a field variable, Eq. (3) can be multiplied by δθ and then in the limit it becomes \begin{equation} \dfrac{dP}{dt}=0,\ {\rm where}\ P=\dfrac{1}{\rho}{\bm{\varsigma}}\cdot\nabla\theta . \end{equation}

This discussion follows that of (Rossby, 1940) and the name potential vorticity derives from his original perspective

based on Eq. (3). He used it in the form \begin{equation} \dfrac{\varsigma_{n}}{\rho\delta h}=\dfrac{f_0+\xi_{\rm PV}}{\rho_0\delta h_0} \end{equation} to give what the value of the relative vorticity would be if the air was taken to a standard latitude (f₀), tilted to the vertical, and the density and isentropic separation (or static stability) changed to standard values denoted by the suffix zero. He then referred to this relative vorticity, \(\xi_\rm PV\), as the potential vorticity, in a manner analogous to potential temperature. The name subsequently became associated with the quantity P. However both Eq. (3) and Eq. (4) refer to a combination of dynamics and thermodynamics. P could equally well be referred to as Potential Stratification. Of course it would be confusing now to do other than use the name PV. However it should always be recalled that PV conservation contains both thermodynamics and dynamics.

Referring again to the form in Eq. (3), and to Fig. 2, the stretching of vorticity is a situation in which δ h and \(\varsigma_n\) both increase. When the isentropic surfaces and the cylinder between them change their orientation, \(\varsigma_n\) keeps the same value but applies to the rotated direction——this is the tilting of vorticity.

Figure 2.  Schematic of (a) a cylinder between two isentropic surfaces, (b) vortex stretching and isentropic surface separation, and (c) vortex and isentropic surface tilting. (Hoskins, 1997)

In the absence of diabatic and frictional processes, both P and θ are conserved in 3-D motion. Therefore P is conserved in 2-D motion on θ-surfaces. Similarly θ is conserved in 2-D motion on P-surfaces. Both perspectives will turn out to be useful.

The standard theory for small Rossby number or large Richardson number, quasi-geostrophic (QG) theory, involves the conservation in horizontal geostrophic motion of the quantity called QGPV, q. However q is not a direct approximation to P. In fact if we define a zeroth order approximation to P which is a function of z only: $$ P_0(z)=\left(\dfrac{f_0}{\rho_r(z)}\right)\dfrac{d\theta_r(z)}{dz} . $$ Then it can be shown that \begin{equation} \left(\dfrac{\partial}{\partial t}\right)_\theta P\approx P_0\dfrac{\partial q}{\partial t}\quad {\rm and}\quad \nabla_\theta P\approx P_0\nabla q . \end{equation} Therefore the rate of change and horizontal advection of q in z-coordinates mimics the same quantities for PV in θ-coordinates. In particular, the material conservation of P on isentropic surfaces is approximated by that of q in horizontal motion. Further, defining P₀ to be a zeroth order approximation to P in θ-coordinates and setting P=P₀+P₁, then it is clear that it is the behaviour of P₁/P₀ in θ-coordinates that is most closely mimicked by the behaviour of q in z-coordinates. In contrast, semi-geostrophic theory uses approximations to PV and its full 3D advection. For further details, see (Hoskins, 1975).

The focus in this paper is on the implications of the material conservation of PV. However heat sources and sinks change the PV on a parcel in a known manner. In the presence of strong latent heat release, PV can change significantly on a time-scale of a day or so. In particular, this can lead to static stabilisation/destabilisation and the creation of high/low values of PV below/above a heating region. Many papers (e.g. Massacand et al., 2001) discuss the importance of this for middle latitude weather systems.

PV plays a fundamental role in the stability of a flow that is independent of one coordinate direction, so-called symmetric stability. In Fig. 3 it is assumed that the flow is independent of the y-coordinate. M=fx+v has gradient in the x, z-plane \((M_x,M_z)=(f+v_x,v_z)=(\varsigma^(z),-\varsigma^(x))\) (here the suffices refer to the directions of the components). Therefore, M-surfaces are in the direction of the absolute vorticity. If a parcel is displaced as in Fig. 3a then its inertial stability means that it accelerates in the horizontal back towards the M-surface. Similarly, the gravitational stability means that it accelerates in the vertical back towards its θ-surface. In this situation, therefore the parcel tends to return from its displaced position. However if the θ-surfaces are more vertical than the M-surfaces, as in Fig. 3b, then these two stabilities act to increase the particle displacement, indicating instability of the basic flow.

Figure 3.  Atmospheric particle displacements and restoring forces in a two-dimensional flow. (a) The usual situation with M-surfaces more vertical than θ-surfaces. (b) θ-surfaces more vertical than M-surfaces, with restoring forces in the horizontal and vertical leading to amplification of the displacement.

It is easily seen that for this 2D flow: \begin{equation} \label{eq2} P\propto J(M,\theta) , \end{equation} and symmetric instability corresponds to fP<0. A full perturbation analysis (Hoskins, 1974) confirms this qualitative parcel discussion and gives the results that follow.

Defining σ_max and \(\sigma_\min\) to be the maximum and minimum frequencies for oscillations to a 2D basic state, then \begin{equation} \label{eq3} \sigma_{\max}^2\times\sigma_{\min}^2\propto fP . \end{equation} From Eq. (4), material conservation of P in the absence of diabatic and frictional processes means that in such a situation an atmosphere that is initially symmetrically stable must remain so.

If the hydrostatic approximation is made, the maximum frequency is for perturbations in the vertical and the minimum frequency is for motion along isentropes. Instability then corresponds to inertial instability along isentropes.

These symmetric stability considerations help an understanding of frontal circulations and frontogenesis, as illustrated in Fig. 4. When there is near surface frontogenesis but the gradients are still weak (Fig. 4a), positive PV means that the M-surfaces are much more vertical than the θ-surfaces and there is a broad frontal circulation. However when the vorticity and gradients of M and θ become large (Fig. 4b) then Eq. (3) implies that the angle between M and θ-surfaces must become small, the maximum frequency becomes large (because of the large stratification in the front). Then Eq. (4) gives that the minimum frequency must becomes small, i.e. there is little inertial stability to motions close to the isentropic slope. The frontal circulation becomes strong and almost along isentropes in the region of strong gradients near the ground. This means that the stretching of vorticity and the creation of strong gradients in M and θ near the ground become even stronger. This is the nonlinear frontogenesis that leads to tendency for frontal discontinuities in v and θ at the surface in a finite time.

Figure 4.  Frontal circulations and M-and θ-surfaces: (a) a weak front, (b) a strong front and (c) an upper tropospheric front.

Figure 4c shows a situation in which there is forcing of frontal circulation in the region of a tropopause jump with θ-surfaces crossing the tropopause. The stability of the large PV stratosphere to motions across θ-surfaces is large. However its stability to motions along isentropes is no larger than in the troposphere. Hence the tendency to increase horizontal temperature gradients in the upper troposphere results in a frontal circulation that can produce tongues of stratospheric air descending down isentropes——the origin of upper tropospheric frontal structure.

As discussed in detail in HMR, invertibility is a vitally important aspect of PV theory. For a flow in balance associated with the rapid rotation of the planet, if the PV distribution is known everywhere on θ-surfaces then, subject to suitable boundary conditions and a knowledge of the total mass between isentropic surfaces, all details of the balanced flow can be determined. It should be recalled that PV gives information on only the local normal component of vorticity and separation of isentropes. However the global knowledge of PV and the balance condition allow all variables to be determined.

A sketch of the typical flow field and isentropes for a positive PV anomaly is shown in Fig. 5a. Recalling Eq. (3), inside the anomaly the large PV is associated with both cyclonic relative vorticity (large absolute vorticity, \(\varsigma\)) and large stratification (small δ h). Above and below, where the PV is not anomalous, small stratification (large δ h) goes with cyclonic relative vorticity (large \(\varsigma\)). To the sides of the anomaly, the large stratification (small δ h) goes with anticyclonic vorticity (small \(\varsigma\)). This implies a maximum in the horizontal flow around the edge of the anomaly.

Figure 5.  Inversions of PV anomalies: (a) a local PV anomaly in a uniform atmosphere, (b) a positive θ region on the tropopause between a high PV stratosphere and a low PV troposphere, and (c) a positive θ region at the surface. In (b) and (c), the situation is circularly symmetric, the horizontal distance shown is 2500 km and the contour intervals are 5 K in θ and 3 m s^-1 in the wind into the section. (b) and (c) are taken from Thorpe (1985).

For the simple low PV troposphere and high PV stratosphere investigated by (Thorpe, 1985) shown in Fig. 5b, a minimum in θ on the tropopause separating them implies a positive PV anomaly on isentropes that cross the tropopause. The tropopause is lowered in this region and there is cyclonic circulation about it. Much as in the simple anomaly the isentropes in the troposphere below are seemingly sucked towards the positive PV anomaly and the cyclonic circulation extends into the region.

Figure 5c shows that a warm anomaly on the lower boundary acts in a similar way. The free troposphere isentropes are "sucked" into the lower boundary and the low stratification (large δ h) in this region of zero PV anomaly goes with cyclonic circulation (large \(\varsigma\)). A cold anomaly on a horizontal upper boundary produces a similar result.

Similarly, anticyclonic circulation goes with a negative PV anomaly, a warm anomaly on a tropopause or a rigid upper boundary, and a cold anomaly on a lower boundary. In this case, the troposphere will in general have large stratification.

At the level of QG theory it is not necessary to know the vertical velocity, w, in order to advect the QGPV. However the vertical motion field is of interest for diagnosis and because of condensation occurring in regions of significant ascent. There are many forms of the diagnostic equation for w, the "omega" equation (see e.g. Hoskins et al., 1978). However it is interesting in the PV perspective to consider another form (Hoskins et al., 2003). Consider an infinitesimally small positive PV anomaly, a δ-function (Fig. 6). The PV anomaly is advected with the flow, and so the isentropes and cyclonic circulation are steady with respect to it. In Fig. 6, the PV anomaly is embedded in a shear flow which is taken to be zero at its level. The steadiness of the isentropes means that the air above the anomaly must flow down the isentrope to the west and up the isentrope to the east. Similarly the air below must flow up the isentrope to the east and down the isentrope to the west. Also, associated with the shear flow, the isentropes will rise to the north so that the cyclonic circulation implies there must be descent, down this isentrope, to the west and ascent, up it, to the east. Adding these components together, the steady state requires "isentropic upglide", w _IU, with ascent ahead (to the east) of the positive PV δ-function anomaly and descent behind (to the west).

Figure 6.  Vertical motion associated with a δ-function positive PV anomaly in a vertically sheared westerly basic flow. The inversion of the PV anomaly gives the isentropes and meridional wind indicated. Since the solution is steady in the frame of reference shown, the air must move up and down the isentropes as indicated. (Hoskins et al., 2003)

In more usual situations that are not steady in any frame of reference, it is found that the isentropic upglide still tends to dominate the field of vertical motion. However if we write $$ w=w_{\rm IU}+w_{\rm ID} , $$ then the additional term, w _ID, is associated with changes in the isentropic structure, isentropic displacement (ID). w _ID can be obtained as the solution of an omega equation with a source term \(-f\frac\partial\partial z\left.\frac\partial q\partial t\right|_c\) in the chosen frame of reference (indicated by the suffix c), and with a boundary condition \(w_\rm ID=-\left.\frac1d\theta_r/dz\frac\partial\theta\partial t\right|_c\). It is the changing PV and boundary θ that lead to isentropic displacement and to the extra component w _ID.

The basic nature of Rossby waves can be seen by considering the situation in Fig. 7 in which there is a basic state with PV increasing towards the north and a localised southward displacement of the PV contours. This gives a localised positive PV perturbation. The associated cyclonic circulation extends beyond the anomaly, and advects the PV contours southwards on the western side and northwards on the eastern side. This tends to create a positive PV perturbation on the western side of the original anomaly, which extends it to the west, and a negative anomaly to the east, giving the start of a wave packet there. These correspond, respectively, to the westerly phase speed and easterly group velocity of Rossby waves relative to a basic flow.

In the presence of a westerly basic flow, there is the possibility of having stationary (i.e. zero phase speed) Rossby waves that will be preferentially forced by mountains and stationary regions of heating. Their eastward group velocity will be larger than the westerly flow, so that wave trains of alternately signed anomalies will occur downstream, to the east, of the forcing. (Yeh, 1949) was the first to analyse this downstream development behaviour.

The situation in Fig. 7 could also refer to potential temperature contours with the values decreasing towards the pole and a cold cyclonic anomaly. Then the same argument as before would apply: these upper boundary Rossby waves would have a westward phase speed and an eastward group velocity relative to the basic flow. However on a lower boundary such a cold anomaly would be anticyclonic and the directions would be reversed: eastwards phase speed and westwards group velocity relative to the basic flow.

Figure 7.  A schematic that shows Rossby wave phase speed and group velocity concepts arising from a single potential vorticity anomaly on a meridional PV gradient (indicated by the PV contours). The arrows show the meridional wind induced by the anomaly.

For more complex geometry and basic flows, a surprisingly accurate theoretical description can be obtained by considering the waves to be present in a slowly varying background. On a sphere great circle ray paths replace straight lines on the plane. Flow variations refract the ray paths and strong jets can act as wave guides (see e.g. Hoskins and Karoly, 1981, and Hoskins and Ambrizzi, 1993).

Baroclinic and barotropic instability can both be simply described in terms of couple Rossby waves (Heifitz et al., 2004). In the situation in Fig. 8 there is a vertical shear in a basic westerly flow. At the upper boundary Rossby waves move eastwards less rapidly than the strong westerly wind, and at the surface they move eastwards more rapidly than the weak flow. Therefore there is the possibility that the waves can move eastwards at similar speeds and interact strongly for a period. For the relative phase of the waves shown in Fig. 8, the northward flow between the upper cold cyclonic anomaly and the warm anticyclonic anomaly extends in the vertical and acts to bring more warm air into the region of the surface warm cyclonic anomaly, and therefore amplifies it. In fact the whole surface wave amplifies. A similar argument gives that the flow associated with the lower boundary wave is such as to amplify the upper boundary wave. Therefore the waves amplify each other by their interaction. It can also be shown that the interaction between the waves acts to help phase-locking between them. If the individual phase speeds of the waves are similar enough and their interaction is strong enough then the waves can be locked together with a relative phase difference that gives mutual amplification. This is a growing normal mode as described in baroclinic instability theory. The coupled Rossby wave perspective gives insight into the essential nature of baroclinic instability. Instead of being on an upper horizontal boundary, the upper wave can be on a positive interior potential vorticity gradient and/or thermal anomalies on the tropopause. Barotropic instability is described by the same argument but with the meridional direction replacing the vertical direction and the waves being on opposite signed PV gradients, implying a PV extremum between them.

Figure 8.  A schematic of the coupling of Rossby waves leading to amplification. In a basic shear flow there are Rossby waves on the upper and lower boundaries with extrema indicated by a + for cyclonic (warm on the lower boundary and cold on the upper boundary) and - for anticyclonic (cold on the lower and warm on the upper boundaries). The meridional flow induced by the upper wave penetrates to the lower level and with the phase shown its warm advection amplifies the warm anomaly there. The basic westerly shear flow is indicated in green and the wave flow relative to this by black arrows.

In (Methven et al., 2005a), the normal modes of a realistic flow on the sphere as described by the primitive Equations were analysed from this perspective and using boundary potential temperature anomaly and interior PV anomaly divided by the basic state PV. The latter was seen in Section 2 to be a variable which is mimicked by QGPV. The lower wave was found to be a surface temperature Rossby wave and the upper wave uses the interior PV gradient and also for the longer waves the tropopause potential temperature gradient. It was found in (Methven et al., 2005b) that this coupled Rossby wave perspective was also helpful for understanding the finite amplitude development of baroclinic waves. The surface wave tends to saturate in the occlusion process, but growth away from the surface continues, leading to the domination of the upper wave in the nonlinear regime.

We have already discussed three examples in which isentropes crossing the tropopause are important: in upper fronts, in inverting to give circulation and stratification anomalies, and in allowing tropopause Rossby waves that can couple to give baroclinic instability. (Hoskins, 1991) referred to the region of isentropic space in which this occurs, typically about 270 K to 380 K, as the Middleworld. The region of isentropes below this is the Underworld, in which the isentropes generally have contact with the boundary layer but not the tropopause. Above about 380 K is the Overworld in which the isentropes are wholly in the stratosphere (or higher).

In a generalisation of the situation shown in Fig. 5b, north of the subtropical jet, the PV is generally below 1.5 PVU (1 PVU = 10^-6 K kg^-1 m² s^-1) in the troposphere and above 3 PVU in the stratosphere. Therefore a value such as 2 PVU (hereafter PV2) can be taken to coincide with the tropopause and is often referred to as a dynamical tropopause. In an adiabatic, frictionless atmosphere both PV and θ are conserved, so that in this situation PV is conserved on θ-surfaces and also θ is conserved on PV-surfaces. We can study the atmosphere in depth using PV on many θ-surfaces. However a single chart that exhibits the important structure in the tropopause region is one of θ on PV2. A sequence in time shows the Middleworld behaviour at the tropopause and the evolution of anomalies that when inverted will give structures like those in Fig. 5b.

Figure 9 shows the development of two very different nonlinear baroclinic waves on jets on the sphere, referred to as LC1 and LC2. Figs. 9a and b give the surface temperature and PV on a particular isentropic surface for LC1 and LC2, respectively. Despite the large amplitude waves, the interaction of the warm surface anomaly and the upper troposphere positive PV anomaly in both is like that discussed in Section 4 in the coupled Rossby wave perspective on baroclinic instability. However the two developments are clearly very different. LC1 shows a small cyclonic spiral in PV in the upper trough and a strong meridional extension of the high PV in a developing anticyclonic wave-breaking event in which the waves rapidly decays. In contrast, LC2 is dominated by a large cyclonic spiral in high PV that is leading to an isolated quasi-circular region of high PV that then remains for more than a week. These contrasting upper tropospheric developments are seen even more clearly in the θ on PV2 panels in Figs. 9c and d.

Figure 9.  The structure of two nonlinear baroclinic waves as seen from a PV perspective. The two waves are from basic states on the sphere with the same baroclinic structure but differing barotropic structures and are referred to as LC1 and LC2 (Life-Cycles 1 and 2). (a) and (b) give θ contours on the lowest model level every 5 K for LC1 and LC2, respectively. In (a) the PV on the 300 K surface is indicated by light shading above 1 and heavier shading above 3, and in (b) the 320 K surface and PV values 2 and 5 are used. (c) and (d) show contours of θ on the PV2 surface for LC1 and LC2, respectively, with contours every 5 K between 290 K and 350 K. (a) and (b) are for day 7.25 and are taken from Methven et al. (2005a, b), and (c) and (d) are for day 7 and are taken from Thorncroft et al. (1993). In all cases two wavelengths are shown in a 120^° sector. In (a) and (b) the sector is from the north pole to 15^°N and longitude and latitude lines are drawn every 15^°. In (c) and (d) the sector is from the north pole to 20^°N.

The Northern Hemisphere fields of θ on PV2 for two quite typical consecutive days in January 2014 are given in Fig. 10. These fields are produced from analyses at ECMWF and are archived routinely at the Reading Department of Meteorology, being available through its web site. Note that because the field is dynamically relevant only poleward of the sub-tropical jet, θ is set to 380 K for values above this. The fluid mechanical nature of the atmosphere is apparent from such pictures and many interesting features are seen. Here we note the cyclonic wrapping of the (potentially) cold polar air in the vortex on the east coast of N America (as LC2) and the more anticyclonic extension of the cold polar air in the vortex in the eastern N Atlantic (as LC1). The subtropical air between these two is advected to high latitudes and develops its own anticyclonic circulation that is acting to cut itself off from its source region. This is the origin of a blocking high. Having cut off, it can only disappear through diabatic processes, which in such a region would be mainly radiative and have a time-scale of a week, or by migrating back to its source region. A PV based analysis of blocking is given in many papers, such as Masato et al. (2014).

Figure 10.  θ on PV2 maps for 0600 UTC on 8 and 9 January 2014 based on ECMWF analyses. The Northern Hemisphere is shown on a polar stereographic projection with 0 deg in the 6 o'clock position.

In the deep tropics where horizontal temperature gradients and variations in stratification are typically small, then consideration of PV is often equivalent to consideration of the vertical component of vorticity and this is the basis of many discussions of tropical systems. However it sometimes enables more insight to consider PV itself.

Figure 11 shows two pictures of PV on the 370 K isentropic surface, which is near 100 hPa and in the upper troposphere in the tropics. Figure11a is the average for August 2009. Above the heating in the Asian Summer Monsoon, the PV values are reduced and these low values give the Monsoon Anticyclone, which drifts west as a Rossby wave. High PV air is drawn from higher latitudes around the eastern side of the low PV by the anticyclonic circulation, forming the mid-Pacific trough and a high PV region on the equatorward side of the anticyclone. The tendency of the anticyclone to split into two has been described by (Hsu and Plumb, 2000) and (Liu et al., 2007) as a barotropic instability of the low PV strip with both signs in PV gradient on its flanks (as discussed in Section 3).

Figure 11.  370 K PV maps for August 2009. (a) average for the month, and (b) 0000 UTC 1 August 2009. The domain is from 60^°W to 180^°E and from 30^°S to 30^°N. (Figures produced by Ricardo Fonseca.)

It is clear from Fig. 11b that at any instant there is a lot of structure in the PV field. This is particularly evident in the high PV air that has been advected around the anticyclone. Also evident are streamers of Northern Hemisphere PV air which have moved deep into the Southern Hemisphere where the ambient PV has the opposite sign. One particular example is near 75^°E. The implication of such pictures for our understanding of the Hadley Cell is the topic of current research.

The intent of this article has been to highlight many of the properties of PV and give an outline of the PV perspective on atmospheric flow. It is a personal account, drawn mainly from the author's own research: no attempt has been made to perform a thorough review of the many relevant and excellent papers on the subject.