Potential Vorticity and the PV Perspective

Brian HOSKINS∗1,

1Department of Meteorology,University of Reading,UK

2Grantham Institute for Climate Change,Imperial College London,UK

Potential Vorticity and the PV Perspective

Brian HOSKINS∗1,2

1Department of Meteorology,University of Reading,UK

2Grantham Institute for Climate Change,Imperial College London,UK

This paper highlights some theoretical aspects of potential vorticity(PV)and discusses some of the insights the PV perspective has given us.The topics covered include the nature of PV,its controlling role in the symmetric stability of the atmosphere,its inversion to give the fl w field Rossby waves and their coupling to give baroclinic instability,PV and midlatitude weather systems and,finall,insights into tropical motions.

circulation,stability,Rossby waves,tropopause

1. Introduction

Since its introduction by Rossby(1940)and,in its full hydrodynamic form,by Ertel(1942),potential vorticity(PV) has figure to a greater or lesser extent in discussions of the dynamics of the atmosphere and ocean.Today,the PV perspective is a majorweapon in the armoury ofatmospheric 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.

2. The concept of PV

The nature of the material conservation of PV can be understood through consideration of conserved quantities for an infinitesimall 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:

Here,ςnis the componentofvorticity normalto the isentropicsurfaces.

Mass conservation for the cylinder gives:

Dividing Eq.(1)by Eq.(2)then gives:

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 fiel variable,Eq.(3)can be multiplied byδθand then in the limit it becomes

This discussion follows that of Rossby(1940)and the name potential vorticity derives from his original perspectivebased on Eq.(3).He used it in the form

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 suffi zero. He then referred to this relative vorticity,ξPV,as the potential vorticity,in a manner analogous to potential temperature. The name subsequently became associated with the quantityP.However both Eq.(3)and Eq.(4)refer to a combination of dynamics and thermodynamics.Pcould 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 whichδhandςnboth increase.When the isentropic surfaces and the cylinder between them change their orientation,ςnkeeps the same value but applies to the rotated direction—this is the tilting of vorticity.

In the absence of diabatic and frictional processes,bothPandθare conserved in 3-D motion.ThereforePis conserved in 2-D motion onθ-surfaces.Similarlyθis conserved in 2-D motion onP-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.Howeverqis not a direct approximation toP.In fact if we defin a zeroth orderapproximation toPwhich is a function ofzonly:

Then it can be shown that

Therefore the rate of change and horizontal advection ofqinz-coordinates mimics the same quantities for PV inθcoordinates.In particular,the material conservation ofPon isentropic surfaces is approximated by that ofqin horizontal motion.Further,defininP0to be a zeroth order approximation toPinθ-coordinates and settingP=P0+P1,then it is clear that it is the behaviour ofP1/P0inθ-coordinates that is most closely mimicked by the behaviour ofqin zcoordinates.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 PVon a parcelin a known manner.In the presence of strong latent heat release,PV can change significantl 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.

3. Symmetric stability and fronts

PV plays a fundamental role in the stability of a f ow that is independent of one coordinate direction,so-called symmetric stability.In Fig.3 it is assumed that the fl w is independent of they-coordinate.M=fx+vhas gradient in thex,z-plane(Mx,Mz)=(f+vx,vz)=(ς(z),-ς(x))(here the suffice 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 stabilitymeans that it accelerates in the horizontal back towards theM-surface.Similarly,the gravitational stability means that it accelerates in the vertical back towards itsθ-surface.In this situation,therefore the parcel tends to return from its displaced position.However if theθ-surfaces are more vertical than theM-surfaces,as in Fig.3b,then these two stabilities act to increase the particle displacement,indicating instability of the basic fl w.

It is easily seen that for this 2D fl w:

and symmetric instability corresponds tofP＜0.A full perturbation analysis(Hoskins,1974)confirm this qualitative parcel discussion and gives the results that follow.

Defininσmaxandσminto be the maximum and minimum frequencies for oscillations to a 2D basic state,then

From Eq.(8),material conservation ofPin 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 theM-surfaces are much more vertical than theθ-surfaces and there is a broad frontal circulation.However when the vorticity and gradients ofMandθbecome large(Fig.4b) then Eq.(7)implies that the angle betweenMandθ-surfaces must become small,the maximum frequency becomes large (because of the large stratificatio in the front).Then Eq.(8) 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 ofvorticity and the creation of strong gradients inMandθnear the ground become even stronger.This is the nonlinear frontogenesis that leads to tendency for frontal discontinuities invandθat the surface in a finit time.

Figure 4c shows a situation in which there is forcing of frontal circulation in the region of a tropopause jump withθ-surfaces crossing the tropopause.The stability of the large PV stratosphere to motions acrossθ-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.

4. Inversion

As discussed in detail in HMR,invertibility is a vitally important aspect of PV theory.For a f ow in balance associated with the rapid rotation of the planet,if the PV distribution is known everywhere onθ-surfaces then,subject to suitable boundary conditions and a knowledge of the total mass between isentropic surfaces,all details of the balanced fl w 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 fl w fiel 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,ς)and large stratificatio (smallδh).Above and below,where the PV is not anomalous,small stratificatio(largeδh)goes with cyclonic relative vorticity(largeς).To the sides of the anomaly, the large stratificatio(smallδh)goes with anticyclonic vorticity(smallς).This implies a maximum in the horizontal fl w around the edge of the anomaly.

For the simple low PV troposphere and high PV stratosphere investigated by Thorpe(1985)shown in Fig.5b,a minimum inθon 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 lowerboundary and the low stratificatio (largeδh)in this region of zero PV anomaly goes with cyclonic circulation(largeς).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 stratifica tion.

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 fiel is of interest for diagnosis and because of condensation occurring in regions of significan ascent.There are many forms of the diagnostic equation forw, 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).Consideran infinitesimall small positive PV anomaly,aδ-function(Fig.6).The PV anomaly is advected with the fl w,and so the isentropes and cyclonic circulation are steady with respect to it.In Fig.6,the PV anomaly is embedded in a shear fl w which is taken to be zero at its level.The steadiness of the isentropes means that the air above the anomaly must fl w down the isentrope to the west and up the isentrope to the east.Similarly the air below must fl w up the isentrope to the east and down the isentrope to the west.Also,associated with the shear fl w,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”,wIU, with ascent ahead(to the east)of the positive PVδ-function anomaly and descent behind(to the west).

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 fiel of vertical motion.However if we write

5. Rossby waves and instability

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 fl w.

In the presence of a westerly basic fl w,there is the possibility of having stationary(i.e.zero phase speed)Rossby waves that will be preferentially forced by mountains and stationary regions ofheating.Theireastward group velocity will be larger than the westerly fl w,so that wave trains of alternately signed anomalies will occur downstream,to the east, of the forcing.Yeh(1949)was the firs 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 upperboundary Rossby waves would have a westward phase speed and an eastward group velocity relative to the basic fl w.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 fl w.

For more complex geometry and basic fl ws,a surprisingly accurate theoreticaldescription 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(Heifit et al., 2004).In the situation in Fig.8 there is a vertical shear in a basic westerly fl w.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 fl w.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 fl w 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 amplifie it. In fact the whole surface wave amplifies A similar argument gives that the fl w 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.

In Methven et al.(2005a),the normal modes of a realistic fl w 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 fi nite amplitude developmentof 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.

6. Mid-latitude weather systems andθon the dynamic tropopause

We have already discussed three examples in which isentropes crossing the tropopause are important:in upper fronts, in inverting to give circulation and stratificatio 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 ofthe situation shown in Fig.5b,north of the subtropical jet,the PV is generally below 1.5 PVU(1PVU=10-6K kg-1m2s-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 andθare conserved,so that in this situation PV is conserved onθ-surfaces and alsoθis conserved on PV-surfaces.We can study the atmosphere in depth using PV on manyθ-surfaces.However a single chart that exhibits the important structure in the tropopause region is one ofθon 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 theθon PV2 panels in Figs.9c and d.

The Northern Hemisphere field ofθon PV2 for two quite typical consecutive days in January 2014 are given in Fig.10.These field 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 fiel is dynamically relevant only poleward of the sub-tropical jet,θis set to 380 K for values above this.The flui 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).

7. Tropical systems

In the deep tropics where horizontal temperature gradients and variations in stratificatio 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.HighPV air is drawn from higher latitudes around the eastern side of the low PV by the anticyclonic circulation,forming the mid-Pacifi 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 gradienton its flank(as discussed in Section 3).

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 ambientPVhas the opposite sign.One particularexample is near 75°E.The implication of such pictures for our understanding of the Hadley Cell is the topic of current research.

8. Concluding comments

The intent of this article has been to highlightmany of the properties of PV and give an outline of the PV perspective on atmospheric fl w.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 relevantand excellent papers on the subject.

Acknowledgements.The material presented has benefitte immensely from discussions with many colleagues over the years and I thank them all.

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(Received 18 July 2014;revised 18 September 2014;accepted 22 September 2014)

∗Corresponding author:Brian HOSKINS

Email:b.hoskins@imperial.ac.uk