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Vortex–Vortex Interactions in the Winter Stratosphere

R. K. SCOTT

Northwest Research Associates, Bellevue, Washington

D. G. DRITSCHEL

School of Mathematics, University of St Andrews, Fife, Scotland

(Manuscript received 14 May 2005, in final form 1 July 2005)

ABSTRACT

This paper examines the interaction of oppositely signed vortices in the compressible (non-Boussinesq)quasigeostrophic system, with a view to understanding vortex interactions in the polar winter stratosphere.A series of simplifying approximations leads to a two-vortex system whose dynamical properties are de-termined principally by two parameters: the ratio of the circulation of the vortices and the vertical sepa-ration of their centroids. For each point in this two-dimensional parameter space a family of equilibriumsolutions exists, further parameterized by the horizontal separation of the vortex centroids, which are stablefor horizontal separations greater than a critical value. The stable equilibria are characterized by vortexdeformations that generally involve stronger deformations of the larger and/or lower of the two vortices.For smaller horizontal separations, the equilibria are unstable and a strongly nonlinear, time-dependentinteraction takes place, typically involving the shedding of material from the larger vortex while the smallervortex remains coherent. Qualitatively, the interactions resemble previous observations of certain strato-spheric sudden warmings that involved the interaction of a growing anticyclonic circulation with the cyclonicpolar vortex.

1. Introduction

The stratospheric sudden warming is one of the mostdramatic fluid dynamical phenomena in the terrestrialatmosphere. In the space of a few days the radiativelydriven cyclonic circulation of the winter stratosphericpolar vortex undergoes a rapid deceleration and polartemperatures increase abruptly. In extreme cases, so-called major warmings, the deceleration and tempera-ture increase is such as to reverse the vertical shear andpole-to-equator temperature gradients throughoutmuch of the stratosphere: zonal mean westerlies asstrong as 80 m s�1 are replaced with easterlies and tem-peratures rise by several tens of degrees.

Traditionally, stratospheric sudden warmings areconsidered to result from the saturation and breakingof planetary waves, which propagate upward from atropospheric source on the steep potential vorticity gra-dients around the polar vortex. This paradigm, based

on a decomposition of the flow into wave and zonalmean components, was demonstrated in the pioneeringmechanistic modeling work of Matsuno (1971) andHolton (1976) and has proved extremely useful, both indescribing the dynamics of sudden warmings them-selves and in quantifying the residual circulation re-sponsible for long-term transport. In this view, majorwarmings typically fall into one of two categories, de-pending on whether the polar vortex is displaced off thepole (wave 1) or split (wave 2). Particularly dramaticexamples of the latter include the warming of February1979 (e.g., McIntyre and Palmer 1983) and the warmingof September 2002 (e.g., Newman and Nash 2005), theonly example of a Southern Hemispheric major warm-ing on record. Recently, Esler and Scott (2005) showedthat resonant excitation of a nonpropagating barotropicmode may provide the dominant forcing mechanism ofthese wave-2 sudden warmings.

Despite the success of the wave–mean flow descrip-tion, the dynamics of the winter stratosphere oftentakes on a more local character. At these times, a de-scription based on interactions of coherent structures ofpotential vorticity (PV) appears more natural than one

Corresponding author address: R. K. Scott, Northwest ResearchAssociates, P.O. Box 3027, Bellevue, WA 98009-3027.E-mail: [email protected]

726 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63

© 2006 American Meteorological Society

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based on wave–mean flow interactions. This synopticview was advocated by O’Neill and Pope (1988), whodescribed the stratospheric evolution in terms of theinteraction of vortices in the presence of topographicforcing in a numerical simulation of a sudden warming.A particularly striking example of vortex merger in thestratosphere was shown in Lahoz et al. (1996, their Fig.16) during the Southern Hemisphere (SH) spring of1994, when a traveling anticyclone merged with a quasi-stationary anticyclone in midlatitudes equatorward ofthe polar vortex edge. The interaction of the resultinganticyclone (also strengthening due to radiative effects)with the radiatively weakening polar vortex contributedto the ensuing final warming, when polar air was en-trained into lower latitudes and the cyclonic circulationwas replaced with anticyclonic circulation in the tran-sition to summer easterlies.

Observations of the Northern Hemisphere winterstratospheric polar vortex reveal a persistent anticy-clonic circulation and associated quasi-stationary ridgeknown as the Aleutian high located near 170°W on theequatorward side of the polar vortex edge (see, e.g.,Harvey and Hitchman 1996, for a climatology). In theSouthern Hemisphere, anticyclones are typically trav-eling during mid- and late winter, but may becomequasi stationary in spring prior to the final warming(Lahoz et al. 1996). Using observations from the UpperAtmosphere Research Satellite (UARS) and the U.K.Met Office (UKMO) unified model, O’Neill et al.(1994) identified the merger of an eastward-travelinganticyclonic vortex with the Aleutian high prior to thesudden warming of January 1992. The interaction of theresulting anticyclonic circulation with the polar vortexcontributed to the sudden warming a few days later.Various other studies (e.g., O’Neill and Pope 1988;O’Neill et al. 1994; Rosier and Lawrence 1999) havelinked the evolution of the Aleutian high to the onset ofstratospheric sudden warmings. Even when the polarvortex does not extensively break down, the interactionof the anticyclonic Aleutian high and the cyclonic polarvortex leads to entrainment of polar and midlatitude airand can contribute significantly to the mixing and trans-port of chemical species across the polar vortex edge(Lahoz et al. 1994, 1996).

The dynamics of interacting vortices in the winterstratosphere closely resembles the behavior of idealizedvortex patches. Numerical studies of the vortex mergerin the two-dimensional Euler equations date back toChristiansen and Zabusky (1973). More recently,Dritschel (1995) documented the equilibrium configu-rations of general two-dimensional vortices, includingthe case of oppositely signed vortices, and illustratedthe nonlinear, time-dependent evolution of the un-

stable configurations using high-resolution contour dy-namics. Of particular interest here is the result that, fortwo oppositely signed vortices of unequal sizes, thelarger vortex generally exhibits the largest deforma-tions, both at equilibrium and during instability. Thisresult, which has also been observed for the case ofthree-dimensional Boussinesq vortices (Reinaud andDritschel 2002; J. N. Reinaud and D. G. Dritschel 2004,unpublished manuscript), suggests the possibility that arelatively small Aleutian high may have a large effecton the dynamics of the polar vortex. Two further ob-servations are in order: first, as vortex disturbances arelarge and the dynamics is far from linear, a wave–meanflow description is not a good representation of theevolution of vortex interactions; and, second, the timescales of vortex interactions in the stratosphere aremuch shorter than typical radiative time scales.

In this paper, we consider in detail the isolated inter-action of two oppositely signed vortices in a compress-ible (i.e., non-Boussinesq) atmosphere. A main cyclonicvortex represents the polar vortex, while a generallysmaller, anticyclonic vortex represents the Aleutianhigh, or the result of previous anticyclonic vortexmerger. We use an adiabatic technique similar to thatused by Dritschel (1995) to sweep out a family of stableequilibrium solutions, followed by high-resolution nu-merical integrations to follow the three-dimensionalevolution of unstable configurations. Our results drawon earlier work (Scott and Dritschel 2005) that used theGreen’s function of the compressible system to showthat a given PV anomaly has a stronger influence on thecirculation above the level of the anomaly than below.

The structure of the paper is as follows. In section 2,we outline the experimental approach, physical param-eters, and numerical procedures, and in particular theadiabatic technique for finding equilibria and construct-ing the initial conditions. In section 3, we describe prop-erties of the equilibrium solutions including the orien-tation and deformation of the vortices at equilibriumand the dependence on circulation ratio and verticaloffset. In section 4, we show examples of the time-dependent, nonlinear evolution of unstable equilibriaand describe the results of a parameter sweep. In sec-tion 5, we consider the influence of the lower boundary.In section 6, we summarize our results and discuss theirimplications for stratospheric sudden warmings.

2. Approach

As a first approximation, we consider the polar vor-tex and Aleutian high as isolated (uniform) patches ofanomalous cyclonic and anticyclonic potential vorticity,respectively. Although O’Neill and Pope (1988) identi-

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fied an important role for tropospheric wave forcing instratospheric vortex–vortex interactions, we restrict at-tention here to the unforced system to isolate as com-pletely as possible the dynamics of the two-vortex sys-tem. Vortex–vortex interactions in the presence of to-pographic forcing will be considered in a future study.

The evolution of the system is completely governedby a relatively small set of parameters, including thesize, shape, strength, and position of the vortices. Wefocus on varying two of these, namely, the relative cir-culation and vertical offset of the two vortices, whilefixing the others to values representative of the winterstratosphere.

For each set of parameter values, equilibrium solu-tions are constructed using the adiabatic technique de-scribed in Dritschel (1995). Two spheroidal vortices,initially well separated, are pushed slowly together byan externally imposed, two-dimensional, area-pre-serving potential flow. The external flow is weakenough that the two vortices remain in approximateequilibrium as they become closer together, thussweeping out a family of equilibrium solutions as afunction of horizontal separation. The procedure is de-scribed in detail in section 2d below.

For small enough horizontal separation, the equilib-ria lose stability and a rapid interaction takes place. Thenonlinear, time-dependent evolution of the vortex in-teractions is followed using the contour-advective semi-Lagrangian method developed by Dritschel and Am-baum (1997). The computational efficiency of thismethod allows a complete investigation of the two-dimensional parameter space described above, withoutcompromising horizontal resolution.

a. Governing equations

For computational efficiency and for a succinct de-scription of the large-scale, balanced dynamical motion,our model is based on the quasigeostrophic equationsin a compressible (non-Boussinesq) atmosphere. Thissystem is arguably the simplest geophysically relevantmodel of rotating, stratified flow. Since time scales forvortex–vortex interactions are short relative to the ra-diative time scale, we include no diabatic forcing term.

The governing equations are

Dq

Dt�

�q

�t� u · �q � 0, �1a�

�h2� �

1�0

�

�z ��0

f2

N2

��

�z� � q, �1b�

�u, ��

�y,��

�x�, �1c�

where q(x, y, z, t) is the (anomalous) PV, � is the geo-strophic streamfunction, u � (u, �) is the horizontalgeostrophic velocity, 2

h is the horizontal Laplacian, xand y are horizontal coordinates, z � �H log p is anappropriate log pressure vertical coordinate, with ver-tical-scale height H, and 0 � exp(�z/H) is the back-ground density (cf. Pedlosky 1987). Here, f � 2� is thepolar value of the Coriolis parameter, where � � 2�day�1 is the planetary rotation rate, and N is a constantbuoyancy frequency.

b. Numerical details

We use the contour advective semi-Lagrangian algo-rithm developed originally by Dritschel and Ambaum(1997) and extended to cylindrical geometry by Ma-caskill et al. (2003). It solves (1) in a cylindrical domainusing a polar coordinate system (r, , z) that rotatesabout the cylindrical axis r � 0 at the rate � � f/2,where f � 4� corresponds to the polar value of theplanetary rotation rate. Lateral boundary conditionsare free slip, and isothermal boundary conditions (i.e.,�z � 0) are imposed at the horizontal upper and lowerboundaries at z � 0 and z � ZT.

For the individual cases presented below, the equa-tions are discretized using 64 layers in the vertical be-tween z � 0 and z � Z

T� 12 H, where the vertical-scale

height H � 6146 m is representative of the stratosphere.Notionally, the vertical domain extends from theground to near the top of the mesosphere. In the hori-zontal, the streamfunction and velocity fields are calcu-lated on a stretched grid of 96 radial and 192 azimuthalpoints, although the PV itself is first interpolated ontoa grid 4 times finer for more accurate inversion. Thestretched grid concentrates radial grid points near thepole, distributing them uniformly in r1/2 for improvedresolution in the region of interest (providing about 30radial grid points between the origin and a notionalvortex edge at r � 3L

R, where L

R� NH/f � 902 km is

the Rossby deformation radius). The lateral boundaryis located at a distance of Rout � 30 L

Rfrom the pole,

far enough away to have practically no effect on theevolution.

For the parameter sweeps, a resolution of half theabove was used, to permit a sufficiently dense samplingof the two-dimensional parameter space. The lower-resolution calculations gave results very similar to thecorresponding high-resolution ones, particularly for theequilibrium states. During the time-dependent interac-tions, small differences between the low- and high-resolution cases developed, but these were restricted tothe details of vortex filamentation; bulk features suchas the wave activity and reduction in total circulation


were found to be largely insensitive to the resolutionsconsidered.

c. Vortex parameters

As first approximations to the polar vortex and Aleu-tian high we consider spheroidal patches of uniformPV, labeled V1 and V2, respectively (subscripts 1 and 2will also be used to identify physical quantities associ-ated with V1 and V2). The two-vortex system can bedescribed by the horizontal and vertical radii of eachundisturbed vortex, their horizontal and vertical sepa-rations, and the ratio of the PV anomalies. For defi-niteness, we fix the vertical–horizontal aspect ratio ofeach vortex, � � c/a, to a value of 4H/3LR. This valueis representative of the stratospheric polar vortex. Fur-ther, we fix the size of V1 to a representative strato-spheric value, with a total depth of 8 H and horizontalradius of 3 LR.

The relative strengths of the two vortices are deter-mined by their volume and PV anomaly. It turns outthat the nature of the interactions depends most signifi-cantly on the ratio of the product of these quantities,that is, on the ratio of the vortex circulations, definedby �i � qiVi, where i � 1, 2, qi is the PV anomaly, andVi � 4�cia

2i /3 is the volume of the undisturbed vortex.

This allows us to fix q1 � �q2 and vary the ratio �2/�1

through the size of V2, which is determined completelyby c2.

With this reduction, the system depends on the cir-culation ratio �2/�1, the vertical positions z1 and z2, andthe horizontal separation d � [(x2 � x1)

2 � (y2 � y1)2]1/2,

where (xi, yi, zi) is the position of each vortex centroidin a Cartesian coordinate system. Of these, �2/�1, z2 �z1, and z1 are treated as external parameters taking thevalues shown in Table 1 (which also summarizes thechoices of fixed parameters). The horizontal separationis swept out during the equilibrium-finding proceduredescribed in section 2d.

Only two values of z1 are considered because inter-actions depend predominantly on the vertical offset z2

� z1. As shown below, the results show only smallquantitative differences between z1 � 6H and z1 �4.5H. Further, in terms of the stratospheric polar vor-tex, these two values bracket the range of plausibleseparations between the lowermost vortex and theground, with separations of 2H � 12 km and 0.5H � 3km, respectively (since c � 4H). Essentially, for z1 �6H the main dynamical evolution takes place farenough above the lower boundary for boundary effectsto be unimportant. The second value of z1 � 4.5 waschosen as this corresponds to the separation wherelower boundary effects begin to affect the evolution(see Scott and Dritschel 2005, for further details).

d. Equilibria

The equilibrium-finding procedure described inDritschel (1995) and Legras and Dritschel (1993) be-gins with two vortices already at (stable) equilibrium,makes a small change to their horizontal separation,and allows the pair to adjust to a new equilibrium state.Practically, this is achieved by imposing a weak, area-preserving, z-independent, external velocity fieldthroughout the domain, which advects each vortex to-ward the origin. The vortices become more deformedas they drift closer together while remaining in quasiequilibrium.

When the vortices are well separated, an equilibriumsolution is given to a good approximation by two coro-tating spheroids. We therefore consider two initiallyspheroidal vortices whose centroids lie, without loss ofgenerality, on the line y � 0 in the corotating frame ofreference, at the points (�x0, 0). We use a value of x0 �Rout/�3 to approximately minimize the combined in-fluence of the circulation due to the image (in the cy-lindrical boundary) of the vortex and the circulationdue to the opposite vortex. As the vortices are broughttogether, the rotation rate of the reference frame isadjusted to follow the vortex corotation, thereby ensur-ing that the vortex centroids remain on the line y � 0 atall times.

In this frame of reference, a suitable external velocityfield is a simple straining flow of the form

�u, �� x, y�, �2�

whose nondimensional magnitude � � 0.05 has beenchosen by experimentation to be small enough to allowthe vortices to adjust to equilibrium as they approacheach other. One benefit of the form (2) is that the ex-ternal flow becomes progressively weaker as the vorti-ces approach each other and become more significantly

TABLE 1. Fixed and varying parameters for the undisturbed po-lar vortex V1 and Aleutian anticyclone V2 in the two-vortex model:PV anomaly q; vertical–horizontal aspect ratio �; vertical semiaxisc (in units of H ); circulation ratio �2/�1; vertical offsetz2 � z1, and polar vortex vertical centroid z1 (both in units of H ).

Fixed V1 V2

q 1 �1� � c/a 4/3 4/3c 4 4|�2/�1|1/3

Varying��2/�1 0.02, 0.04, . . . , 0.98, 1z2 � z1 �2, �1.9, . . . , 1.9, 2z1 6, 4.5


deformed. A family of stable equilibrium solutions arethus swept out over a range of horizontal separationsuntil the separation becomes small enough that the sys-tem loses stability.

e. Time-dependent interaction

We study the time-dependent evolution of the insta-bility by solving the full equations (1) with initial con-ditions obtained from the above equilibrium-findingprocedure at small horizontal separation. At present,we do not have a precise method for determining ex-actly when the equilibria become unstable. However,an inspection of a large number of cases suggests thatthe vortices are always unstable when their deforma-tion becomes sufficiently large. It turns out that a suit-able measure of vortex deformation is given by the sec-ond and third elliptical moments, M(2)

i and M(3)i , where

Mi�2� �

1Mi

�Vi

�0�z��x2 � y2 � re2�z�� dx dy dz, �3a�

Mi�3� �

1Mi

�Vi

�0�x�y3 dx dy dz, �3b�

and where Mi � �Vi0 dx dy dz is the total mass of the

vortex and re(z) is the mean horizontal radius at eachlevel. We assume a nondimensional scaling of x and ysuch that Rout � 1. Here, M(2)

i measures the horizontalellipticity of the vortex and M(3)

i measures the depar-ture from symmetry in the y axis, which typically takesthe form of a crescent-shaped deformation of the largervortex. In all the cases studied, the system is unstablewhen the combination

M � max�Mi�3�, 0.05Mi

�2�� 4�

exceeds a certain threshold, M c. For unequally sizedvortices the deformation involves larger M(3)

i andsmaller M(2)

i . When the vortices are of similar size, how-ever, the asymmetry in the y axis is small and the de-formation involves mainly M(2)

i .We choose the threshold value M c � 0.0001, which

has been empirically determined to ensure that the sys-tem is well within the margin of instability. For margin-ally unstable vortices the instability is relatively weak,with smaller growth rates and weaker interaction thanoccurs during the merger of like-signed vortices.

3. Equilibrium solutions

We begin by considering the equilibrium solutions oftwo equal volume, vertically aligned, oppositely signedvortices. Obviously this configuration is not intended toresemble the winter stratosphere. Rather, it is useful

because it illustrates the type of vortex deformationsthat occur in the absence of the asymmetries introducedby differences in size and vertical position. Such a con-figuration might conceivably be approached in SHspring before the onset of the final warming, when ra-diative effects weaken the polar vortex and strengthenthe Australian high, but the dipole will translate and isunlikely to remain stable over the pole for long.

As described in section 2d, a family of steady states isswept out as the originally spheroidal vortices areslowly brought together. As the vortex separation de-creases, the vortices deform by flattening in the direc-tion of their alignment, predominantly on the inner-most faces (see Fig. 1). For the extreme case on theright, which is actually within the margin of instability,the faces are almost touching and the top view re-sembles the limiting case for two-dimensional vorticesdescribed in Dritschel (1995, his Fig. 6e).

In addition to the flattening, the vortices also developa vertical tilt with lower vortex levels closer togetherand upper levels farther apart. The tilting appears con-sistent with the vertical dependence of the induced cir-culation of each vortex. The exponential variation ofthe background density profile introduces an anisot-ropy in the quasigeostrophic Green’s function, withslower decay above than below the vortex, whichmeans that the induced circulation of a uniform vortexpatch is stronger above the vortex than below (see Scottand Dritschel 2005, e.g., their Fig. 2). Considering thevortex pair as a stack of two-dimensional dipoles trans-lating in the positive y direction (i.e., toward the top ofFig. 1d), the dipole at upper levels will tend to translatefaster than that at lower levels. This will cause the pairto tilt along the y axis, as observed in the figure. Toreduce the y tilt, the vortices tilt away from each other(in x) with height, thereby increasing the horizontalseparation and reducing the strength of the circulationeach vortex feels from its pair.

We now consider how the symmetry of the aboveequilibria is broken, first, when the vortices are of un-equal sizes, and second, when they are vertically offset.Equilibria for unequal vortices with zero vertical offsetare shown in Fig. 2 for circulation ratios of ��2/�1 �0.8, 0.6, 0.4, and 0.2. The corresponding equal volumecase was shown in Fig. 1d. Note that the equilibriashown are for M just exceeding M c, and represent moreor less the limit of the equilibrium-finding procedure:although these equilibria are unstable, the growth rateof the instability up to these horizontal separations issmall enough that significant departures from equilib-rium have not yet occurred. As for the equal volumecase, the vortices again become flattened toward eachother and tilted away from each other with height. Per-


haps counterintuitively, however, it is the larger vortexthat experiences the greatest deformation (see Fig. 2).Such behavior was identified already by Dritschel(1995) in the two-dimensional case and has also beenobserved in the three-dimensional Boussinesq case(J. N. Reinaud and D. G. Dritschel 2004, unpublished

manuscript). In addition to flattening, the larger vortexalso develops a characteristic concave shape orientedtoward the smaller vortex.

Figure 3 shows the equilibria for the case of equalvolume but vertically offset vortices, again at the pointwhere M just exceeds M c. As the vertical offset is in-

FIG. 1. Equilibrium states for the equal volume, vertically aligned vortex pair for different horizontal separations: (a)–(d) a top viewand (e)–(h) side view perspective from an elevation of 30°. The negative (anticyclonic) vortex is drawn using dashed contours. The axisin (d) indicates the orientation relative to the Cartesian coordinates used in section 2c.

FIG. 2. Equilibrium states for vertically aligned vortex pairs ofunequal volume: (a) circulation ratio ��2/�1 � 0.8, (b) ��2/�1 �0.6, (c) ��2/�1 � 0.4, and (d) ��2/�1 � 0.2. The top view is shownlarge; the inset shows the corresponding side view.

FIG. 3. Equilibrium states for equal volume vortex pairs withdifferent vertical offsets: (a) z2 � z1 � 0.8, (b) z2 � z1 � 1.6, (c)z2 � z1 � �0.8, and (d) z2 � z1 � �1.6. The top view is shownlarge; the inset shows the corresponding side view.


creased, the tilt of the vortices, which persists for smalloffsets, develops a twist, with the vortices partiallywrapped together for z2 � z1 � �1.6H. We find thatnow the lower vortex is always the more deformed ofthe pair. As with the tilt, this property can be under-stood in terms of the slower decay of the Green’s func-tion above the vortex than below: at a given level, thecirculation induced by the lower vortex is on averagestronger than that induced by the upper vortex. Thelower vortex can therefore be regarded in some senseas the stronger of the pair and experiences strongerdeformation as in the case of unequal volume vortices.Note that the sense of the twist here is eastward withheight, in contrast to the occasionally westward tilt withheight of the Aleutian high (Harvey and Hitchman1996), the latter is most likely a signature of upwardwave propagation.

To quantify the combined effects of unequal volumeand nonzero vertical offset on the vortex deformationwe calculate, separately for each vortex, the third mo-ment M(3)

i , where i � 1 and 2 for the cyclonic andanticyclonic vortices, respectively. Figure 4 shows M(3)

1

� M(3)2 for circulation ratios ��2/�1 between 0.05 and 1

and vertical offsets z2 � z1 between �2H and 2H. Asseen above, along the line z2 � z1 � 0 it is always thestrongest vortex that is deformed the most. Similarlyalong the line ��2/�1 � 1 it is always the lower vortexthat is deformed the most. In general, there is a well-defined crossover point in parameter space where theeffects of vortex strength and vertical offset are in exactbalance and the vortices are equally deformed. Aboveand to the left of that line (i.e., for a smaller and higheranticyclone) the cyclonic vortex experiences the mostdeformation.

4. Nonlinear evolution

We next consider the time-dependent evolution ofunstable equilibrium solutions. As described in section2e, we use the criterion M � M c to select unstable initialconditions from which to integrate the equations (1).For all the cases considered, this criterion ensures thatthe equilibria are indeed unstable. Although equilibriaare often unstable at smaller values of M (i.e., at largerhorizontal separation) the growth rate for the instabil-ity only becomes significant (relative to radiative relax-ation rates in the stratosphere) at larger M .

Figure 5 shows an example of the evolution of theinstability for a circulation ratio ��2/�1 � 0.6 and ver-tical offset z2 � z1 � 0. The initial condition in Fig. 5ais the same as the equilibrium solution shown in Fig. 2b.As the instability develops, the cyclonic (polar) vortexbecomes increasingly deformed, wrapping around thesmaller, anticyclonic vortex. By t � 6 days the polarvortex has become so elongated that it wraps up intotwo distinct vortices, the smaller of these forming adipole pair with the anticyclone. In addition to theshedding of a second vortex, the polar vortex also be-comes highly tilted, which results in the subsequentshedding of vortex material at upper levels (notshown). Remarkably, the anticyclonic vortex remainscoherent and approximately ellipsoidal throughout theevolution.

a. Parameter sweep

We use two measures to quantify the total deforma-tion of the polar vortex resulting from the unstable in-teraction; namely, the time-averaged wave activity andthe reduction in the zonal mean zonal velocity. Thetotal wave activity A(t) is the vertical integral of

A�z, t� � ��z�q��z�

�Y�, z, t� � Ye�z��2 d, �5�

which represents the departure of the polar vortex froma circular cross section (Dritschel and Saravanan 1994).Here the integral is taken around a closed contour � (orcollection of contours, if the original contour breaksup) at height z, is an azimuthal coordinate, Y � 1⁄2 r2

(so that dY d is the differential area), Ye � 1⁄2 r2e, and

re is the radius of the undisturbed circular contour en-closing the same area as �. The total wave activity isnormalized by the total angular impulse of the polarvortex at t � 0. Note that here we consider the waveactivity of the polar vortex alone, as a measure of itsdeformation; the combined wave activity of the polarvortex plus anticyclone is conserved in the absence ofdissipation.

FIG. 4. Relative vortex deformation measured by the third mo-ment, M(3)

1 � M(3)2 , where i � 1 and 2 for the cyclonic and anti-

cyclonic vortices, respectively. Positive values are contoured solid,negative values are dashed.


The reduction in zonal mean zonal velocity is definedas [U(0) � U(�)]/U(0), where U(t) � max(r,z)u(r, z, t) isthe domain maximum of the zonal mean zonal velocityat time t. The long-time value U(�) is defined as thetime average from t � 10 to t � 40. In calculating u, wetake the zonal mean to be the azimuthal average rela-tive to the centroid of the polar vortex. While, at firstsight, it is more natural to take the zonal mean relativeto the global potential vorticity centroid, it is clear thatsuch an approach fails for values of ��2/�1 nearingunity, when the global centroid is located increasinglydistant from the region of interest. Our choice is moti-vated by our objective of quantifying the deformationof the polar vortex. Further, for small �2/�1 the differ-ence between the polar vortex centroid and the globalcentroid is small. We note, however, that our choicediffers from the traditional, geographically relativezonal mean used in spherical geometry in one impor-tant respect: on the sphere, the displacement of a vortexoff the pole implies a deceleration of the traditionalzonal mean zonal velocity, even in the absence of othervortex deformations. In contrast, taking the zonal meanrelative to the vortex centroid implies that all zonalmeans are by definition translation invariant.

We calculate these two diagnostics for a series ofsimulations covering the two-dimensional parameter

space defined by ��2/�1 and z2 � z1 taking the valuesshown in Table 1. Since the PV anomalies of each vor-tex are fixed, the range of circulation ratios correspondsto anticyclones of volumes between 1 and 1/20 times thevolume of the polar vortex. The range of values of ver-tical offset allows for any vertical position of the anti-cyclone such that it remains entirely within the verticaldomain. Negative offset values correspond to a loweranticyclone.

Figure 6 shows the results from the case z1 � 6H, forwhich the polar vortex is in the center of the domainwith the lowermost vortex level at a distance 2H fromthe ground. Both diagnostics show that significant in-teraction occurs over a highly localized region in pa-rameter space. This region coincides closely with thetransition region seen in Fig. 4 where both vortices haveapproximately equal levels of deformation. Thus, for��2/�1 � 1, the maximum interaction, or the maximumeffect of the anticyclone on the polar vortex, occurs forconfigurations in which the centroid of the anticycloneis slightly below that of the polar vortex. When theanticyclone is lower there is greater interaction with thelower, and denser, levels of the polar vortex, and hencethere is a larger contribution to wave activity. On theother hand, if the anticyclone is too much lower it ex-periences stronger deformation than the polar vortex.

FIG. 5. Evolution for ��2/�1 � 0.6, z2 � z1 � 0 at t � 0, 2, 4, 6, 8, and 10 days (from upper left to lower right).


b. Dependence on vertical offset

We now examine in more detail the dependence ofthe vortex evolution on the vertical offset. Figure 7shows the evolution at t � 0, 4, 8 days for ��2/�1 � 0.6and values of vertical offset z2 � z1 � 0.4H, �0.4H, and�0.8H. See also the case z2 � z1 � 0.0 shown in Fig. 5.The maximum interaction occurs for z2 � z1 � �0.4H,consistent with Fig. 6a. Although a similar volume ofmaterial is shed from the polar vortex for z2 � z1 � 0,close inspection reveals that for z2 � z1 � �0.4H ma-terial is shed from the vortex at lower levels and, be-cause of density weighting, therefore results in a greaterchange in vortex angular momentum and wave activity.For z2 � z1 � �0.8H, material is shed from the polarvortex at still lower levels (the lowest contours beingcompletely removed), although the volume is smaller.For z2 � z1 � 0.4H (i.e., when the anticyclone is higherthan the polar vortex), only a small volume of material

is shed from the upper vortex levels and the density-weighted contribution to the angular momentum bud-get is small.

The distribution of zonal mean zonal velocity relativeto the polar vortex centroid, u, as a function of radialcoordinate (corresponding to latitude) and height, alsoillustrates the relative strengths of interaction for dif-ferent vertical offsets. Figure 8 shows u at (left column)t � 0 and (right column) t � 10 days for z2 � z1 � 0.4,0, �0.4, �0.6, �0.8, and �1.2H, respectively. At t � 0the polar vortex appears as a broad jet with a maximumbetween r � 2 LR and r � 3 LR and between z � 7Hand z � 8H, roughly corresponding to between 60° and70° latitude and between 42- and 48-km height, notdissimilar to the winter stratospheric polar vortex. De-pending on the height of the anticyclone, the interac-tion appears as a sudden deceleration of the jet withcharacteristics ranging from weak deceleration con-fined to the upper vortex (z2 � z1 � �0.4H) to com-plete reversal of u throughout most of the domain.

In terms of the deceleration of u, the maximum effectof the anticyclone on the polar vortex is obtained forz2 � z1 � �0.6H and �0.8H, where the zonal meanwinds are reduced to around zero or reversed in a ma-jor sudden warming–like interaction. When the anticy-clone is lower than this it has very little effect on thecirculation of the polar vortex. When the anticyclone ishigher its main effect is the reduction of zonal meanwinds in the upper part of the domain only.

If we consider the appropriate scaling of vortexanomalies we find that time scales for the interactionare similar to those of stratospheric sudden warmings.With the chosen values of PV anomalies (�f ), themaximum deceleration of the vortex is usually reachedaround t � 10 days. However, as can be seen from Fig.8, these anomalies are somewhat weak, correspondingto a jet maximum of up to around 35 m s�1, aroundhalf-typical stratospheric values. Doubling the PVanomalies to give more realistic zonal velocities isequivalent to halving the time scale: the same decelera-tion occurs as shown in Fig. 8 but with the contourinterval of 10 m s�1 and over a time interval of 5 insteadof 10 days, consistent with velocity changes and timescales of observed sudden warmings.

c. Dependence on circulation ratio

We now take a closer look at vortex interactions fordifferent �2/�1. First, we consider the case of a verystrong anticyclone, with ��2/�1 � 0.8. Although it ishighly unlikely that a coherent anticyclone of thisstrength could ever develop in the winter stratosphere,it is not inconceivable that such a situation might arisein the Southern Hemisphere stratosphere in spring, im-

FIG. 6. Measures of the reduction in main (cyclonic) vortexintensity resulting from the interaction. (top) Total final waveactivity of the cyclonic vortex normalized by the initial angularimpulse. (bottom) Reduction in the domain maximum of thezonal mean zonal velocity [U(0) � U(�)]/U(0). In both cases thesubscript � means the time average from t � 10 to t � 40 days.Zonal means are taken with respect to the centroid of the cyclonicvortex. Contouring is the same as in Fig. 4.


mediately prior to the final warming, when radiativeconditions are weakening the polar vortex and, in com-bination with dynamical processes, strengthening theAustralian anticyclone. Figure 16 in Lahoz et al. (1996)indicates that anticyclonic circulations approaching thesize of the polar vortex were present in the SouthernHemisphere middle stratosphere in October 1992.

The case of ��2/�1 � 0.8 and z2 � z1 � 0 is shown inFig. 9 at times t � 0, 4, 8, 12, 16, and 20 days. At t � 4the vortex begins to split in two approximately equalparts, these becoming increasingly separated by t � 8.Up until this time the anticyclone remains relativelycoherent and undeformed. At t � 8, however, the an-

ticyclone begins to interact with one of the separatedremnants of the main vortex. Because of their semi-isolation from the other vortex remnant, their interac-tion follows the general pattern for two unequal vorti-ces observed so far. Now, however, the anticyclone isthe larger vortex, and the remnant the smaller, morecoherent vortex, with the result that the anticyclonesplits in two (from t � 12). The four vortices thusformed subsequently pair up and move apart undertheir induced circulation, apparently in a stable quasiequilibrium and with little further interaction.

The case of ��2/�1 � 0.2 is arguably more relevant tothe winter stratospheric case. It illustrates that even a

FIG. 7. Evolution for ��2/�1 � 0.6 at (left to right) t � 0, 4, 8 days: (a) z2 � z1 � 0.4H, (b) z2 � z1 � �0.4H, and(c) z2 � z1 � �0.8H.


relatively small anticyclone can induce a significant po-lar vortex deformation and deceleration of the vortexcirculation if it lies at the appropriate vertical offset.Figures 10a,b show the cases z2 � z1 � �1.2H andz2 � z1 � �2.0H, respectively, at times t � 0, 4, and 8days. For z2 � z1 � �1.2H material is pulled off thepolar vortex from the central–lower levels. For z2 � z1

� �2.0H less material is pulled off, but it consists ofdenser air from lower levels and has a potentially largerimpact on the vortex angular momentum. Figures 11a,bshow the corresponding zonal mean zonal velocity att � 0 and t � 10, respectively. In each case the vortex isdecelerated strongly, resembling a minor warming. Theregions of strongest deceleration are different betweenthe cases: deceleration is confined to the upper vortexfor z2 � z1 � �1.2H, whereas it extends lower for z2 �z1 � �2, 0H. This pattern of deceleration is consistentwith the stronger upward influence of PV anomalies inthe compressible quasigeostrophic system identifiedpreviously by the authors (Scott and Dritschel 2005).

5. Lower boundary influence

The presence of a lower boundary has an importanteffect on the circulation induced by a PV anomaly inthe compressible quasigeostrophic system. As shown inScott and Dritschel (2005) the Green’s function of a PVpoint anomaly close to a lower boundary contains astrong barotropic component with logarithmic horizon-tal dependence, which dominates the response awayfrom the anomaly. In addition, the Green’s functionsingularity at the lower boundary is doubled due to theimage in the boundary, asymptoting to �1⁄2�r for smallr. For the dynamics of a single vortex, boundary effectswere found to become important when the separationbetween the lowermost vortex and the lower boundarywas around 0.5–1 density-scale heights.

Since in the above analysis, the polar vortex is situ-ated at a separation of 2H from the lower boundary, itis important to establish whether the results hold atsmaller separations. Boundary effects will potentiallyalter both the equilibrium solutions and the nonlinearinteractions. We therefore repeated the analysis for z1

� 4.5H, for which the lowermost level of the polarvortex is separated by only 0.5H from the lower bound-ary and boundary effects might be important. Althoughit is interesting to consider even smaller separations,these are less relevant to the stratospheric case, wherethe vortex is generally well separated from the ground.Our choice of z1 � 4.5H already corresponds to smallerseparations than typically observed in the winter atmo-sphere.

It turns out that even at this small separation from

FIG. 8. Zonal mean zonal velocity, defined relative to the main(cyclonic) vortex centroid at (left) t � 0 and (right) t � 10 days, forthe case ��2/�1 � 0.6 and for vertical offsets z1 � z2 � (a) 0.4, (b)0.0, (c) �0.4, (d) �0.6, (e) �0.8, and (f) �1.2H. Dotted contoursdenote negative values and the contour interval is 5 m s�1.


the lower boundary, both equilibrium solutions andnonlinear interactions are qualitatively unchanged, al-though certain systematic differences are observed. Forexample, the equilibrium solutions such as those shownin Figs. 1–3 are in general characterized by a smallervertical tilt, with the vortices in the lowermost levelsless squashed together. This effect is likely due to theintensification of the Green’s function singularity forPV close to the lower boundary. Similarly the lowerlevels of the vortices in general experience less defor-mation for z1 � 4.5H than for z1 � 6H.

Because a greater fraction of the total vortex defor-mation occurs at upper levels for z1 � 4.5H, the mea-sure M will generally take smaller values than for z1 �6H, on account of the density weighting in (3). As aresult, equilibria in the z1 � 4.5H case generally be-come unstable at lower values of M , and it was foundthat a lower-threshold value of M c � 0.00005 was amore suitable indication of when to start the time-dependent evolution of the instability; higher values ofM c resulted in vortices that were clearly deformed wellbeyond their unstable equilibrium configuration.

Results of the time-dependent evolution of the insta-bility are summarized in Fig. 12 for z1 � 4.5 H, corre-sponding to the parameter sweep discussed in section4a and Fig. 6 above. Again, there is a well-defined re-

gion in parameter space over which the strength of theinteraction is maximal. Note that, because of the con-straint that both vortices lie entirely inside the domain,the parameter sweep is truncated at large negative ver-tical offset and circulation ratio; points on the boundaryof the shaded region correspond to cases where theanticyclone exactly touches the lower boundary. Com-parison of several individual cases (not shown) revealsome differences between the cases z1 � 4.5H and z1 �6H, for example, generally weaker tilting of the polarvortex and shedding of vortex material over a deepervertical range for z1 � 4.5H, presumably a result of thestronger barotropic component of the circulation. Themain conclusions, however, are unchanged; namely,that the larger of the two vortices is always the moresignificantly deformed, that a relatively modest anticy-clone can lead to substantial polar vortex deformations,and that the strongest interactions occur over roughlythe same, well-defined region of parameter space.

6. Discussion

Although highly idealized, the above results were de-veloped as a model of vortex interactions in the winterpolar stratosphere. Our intention is to study processesof vortex interaction general enough that they do not

FIG. 9. Evolution for ��2/�1 � 0.8, z2 � z1 � 0, at t � 0, 4, 8, 12, 16, and 20 days (from upper left to lower right; top view).


depend on the finer details of the model, but which areinstead robust and fundamental properties of the fluiddynamical system. In our model, the larger, cyclonicvortex represents the polar vortex and the smaller, an-ticyclonic vortex represents the Aleutian high, or othersimilar anticyclonic circulations often observed in both

the Northern and Southern Hemisphere winter strato-spheres. The approximation of these structures by uni-form, oppositely signed PV anomalies having a sphe-roidal undisturbed state leads to perhaps the simplestconfiguration possible in three dimensions, some mightargue too simple. On the other hand, the zonal meanzonal velocity profile associated with these structures isnot unrealistic (e.g., Fig. 8). Further, given the increas-ingly well-accepted view of the polar vortex edge as aregion of intense PV gradients, the approximation of avortex patch is in some respects better than, for ex-ample, the uniform PV gradient approximations used inpioneering work on wave propagation. We emphasizethat our aim, in common with those early papers, is toinvestigate fundamental processes that might be rel-evant to the winter stratosphere, rather than to repro-duce in detail the stratospheric evolution.

Our motivation for this study stems from the work ofO’Neill and Pope (1988) who identified coherent anti-cyclonic circulations associated with the stationaryAleutian high and eastward-traveling PV anomalies,situated on the equatorward flank of the polar vortex.Although the origin of these anomalies can be consid-ered naturally in the framework of a wave–mean flowdecomposition, their subsequent development re-sembles more closely the evolution of coherent vortical

FIG. 10. Evolution for (a) ��2/�1 � 0.2, z2 � z1 � �1.2H and (b) z2 � z1 � �2.0H at t � 0, 4, and 8 days (left to right).

FIG. 11. Zonal mean zonal velocity, defined relative to the main(cyclonic) vortex centroid at (left) t � 0 and (right) t � 10 days, forthe case ��2/�1 � 0.2 and for vertical offsets (a) z2 � z1 � �1.2H and (b) z2 � z1 � �2.0 H. Contours are the same as in Fig. 8.


structures, involving strongly local and nonlinear dy-namics, for example during the merger of traveling andstationary anticyclones and the interaction of anticy-clones with the polar vortex.

Observational evidence of equilibrium configura-tions consisting of a polar vortex and Aleutian high (orother anticyclonic circulations) has been documented inprevious studies. The polar vortex and Aleutian highoften coexist for relatively long periods without stronginteraction. The cross sections of zonal wind shown inFigs. 5 and 9 of O’Neill et al. (1994) suggests a polarvortex that is tilted away from the weaker circulation ofthe Aleutian high with height, similar to the tilting dis-cussed in section 3. Present meteorological analysesare now of sufficient resolution that accurate three-dimensional images of PV can be constructed through-out the stratosphere. It would be interesting to examinethese datasets for further evidence of quasi-equilibriumstates involving the polar vortex and Aleutian high, andto test the dependence of the morphology of equilib-

rium states predicted here on circulation strength andvertical offset.

The most dramatic instances of vortex–vortex inter-action above occur for substantial circulation ratios,with ��2/�1 �0.6–0.8. Certainly these values appearlarger than anything that could be realistically attainedin the polar stratosphere during winter radiative condi-tions. In the Southern Hemisphere late winter andspring, however, a strong stationary anticyclonic circu-lation, the Australian high, typically develops at thesame time as the polar vortex is weakening through ra-diative processes. A striking example of the merger ofthis stationary anticyclone with an eastward-travelinganticyclone was discussed in Lahoz et al. (1996). Re-constructed maps of PV on the 1100-K isentropic sur-face suggest that the resulting anticyclonic circulationwas indeed of comparable magnitude to the polar vor-tex. The examples of strong interaction shown heremay therefore turn out to be more applicable to thedynamics of the Southern Hemisphere’s final warming.

In general, and as to be expected, smaller circulationratios resulted in less dramatic vortex–vortex interac-tions. However, an interesting result is that for verydifferently sized vortices, it is always the larger one thatexperiences the stronger deformations. That is, even avery weak anticyclonic circulation can induce large de-formations to the polar vortex, and, if at the appropri-ate vertical offset, can result in shedding of vortex ma-terial and mean flow reduction similar to that observedin minor stratospheric warmings.

One important ingredient that might significantly en-hance the potential for a weak anticyclone to inducestrong polar vortex deformations is the presence of ex-ternal forcing due, for example, to topography. O’Neilland Pope (1988) suggested that strong vortex interac-tions in the stratosphere relied on the presence of to-pographic forcing. While a full treatment of the effectof topography is clearly beyond the scope of the presentwork, an idea of its effect can be obtained by consid-ering the unstable evolution of equilibria beginning far-ther within the margin of instability (closer together).This can be practically achieved by following the equi-librium finding procedure beyond the point where thedeformation M exceeds M c. The weak external strain-ing flow that brings the vortices together here is similarto the flow induced by topographic forcing. Underthese conditions the interaction of a relatively weakanticyclone can be substantially enhanced. As an ex-ample, Fig. 13 shows the evolution for case with ��2/�1

� 0.1 and z2 � z1 � 0. While the anticyclone againremains relatively undeformed throughout the interac-tion, the polar vortex is this time violently deformed,

FIG. 12. Same as in Fig. 6, but for polar vortex centroid locatedat z1 � 4.5H. Parameter values are constrained by the require-ment that both vortices lie entirely within the vertical domain.Points on the boundary of the shaded region correspond to caseswhere the anticyclone exactly touches the ground.


becoming strongly sheared in the vertical and eventu-ally splitting in two. A similar shearing of the polarvortex occurred for other vertical offsets. The anticy-clone in this case is half as strong as those presented inthe weak interaction cases above—a mere 10% of thecirculation of the polar vortex. The extent to which theinstability of the two-vortex system manifests itself al-most entirely through the destruction of the muchlarger polar vortex is remarkable.

Acknowledgments. Support for this research was pro-vided by the U.K. Natural Environment ResearchCouncil under Grant NER/B/S/2002/00567.

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FIG. 13. Evolution for ��2/�1 � 0.1, z2 � z1 � 0 at t � 0, 8, and 16 days (left to right) in the presence of persistent external strain.


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