Theor. Appl. Climatol. 76, 189–202 (2003)DOI 10.1007/s00704-003-0011-x

1 The Open University of Israel, Tel-Aviv, Israel2 Department of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty

of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

Rotation of mid-latitude binary cyclones: a potentialvorticity approach

B. Ziv1 and P. Alpert2

With 14 Figures

Received March 11, 2002; revised June 27, 2003; accepted August 6, 2003Published online November 20, 2003 # Springer-Verlag 2003

Summary

Two cyclonic vortices close to each other, a ‘binarycyclone’ or ‘binary system’, tend to rotate cyclonicallyrelative to one another and to merge, i.e. the ‘‘Fujiwharaeffect’’. The point vortex model that represents barotropicbinary cyclones predicts their rotation features as follows.The rotation rate is proportional linearly to the sum of thecyclones’ intensities and inversely to the square of theirseparation distance while the more intense cyclone rotatesslower. Our earlier observational analysis of 1423 mid-latitude binary cyclones (Ziv and Alpert, 1995) showeda reasonable fit to theory, except for the absence of acorrelation between individual speeds and intensities withinthe binary systems, and a reversal of the inverse rotation-separation relationship at the range of 1400–1800 km.

This study is the first attempt to describe the mid-latitudebinary systems using potential vorticity concepts (PV think-ing), which implies that a binary interaction takes placebetween the 3-D flow patterns induced by upper-PV orsurface-thermal anomalies rather than by the surfacecyclones alone. It is argued that the upper-anomalies dom-inate the rotation process, and hence the rotational speeds ofthe interacting surface cyclones are more closely correlatedwith the relative intensities of their corresponding upper-level anomalies rather than with their own intensities, asreflected in weather charts. Data analysis indicates thatmid-latitude binary cyclones are normally associated withat least one upper-PV anomaly. This explains the absenceof a correlation between the rotation speed and the intensityof the surface cyclones there.

A unique type of a mid-latitude binary system is identi-fied, in which one cyclone coincides with an upper majorPV-anomaly and the other moves along the periphery of theformer. Such a binary system is entitled here the ‘ContactBinary System’ (CBS), in contrast with remote interactingsystems implied by the point vortex theory.

Analytical considerations yield an increase in the rotationrate with separation for CBSs of separation smaller than�1000–1500 km, in contrast to the normal decrease with R2.The contribution of CBSs is suggested here to explain theabnormal increase in rotation rate at 1400–1900 km range.

1. Introduction

Two cyclonic vortices in close proximity, i.e.a binary cyclone, tend to rotate cyclonicallyrelative to each other and to eventually merge(Fujiwhara, 1931). This interaction was entitled‘‘the Fujiwhara effect’’ or ‘‘binary interaction’’.The rotational aspect of this interaction wasquantitatively formulated for a barotropic fluidusing the simple point vortex model (e.g. Lamb,1945; Batchelor, 1980; Aref, 1983). According tothis model each vortex, represented by an infi-nitely small core with positive vorticity, inducesa circular flow pattern throughout the fluid. Whenapplied to two cyclonic point vortices, denoted

1 and 2, with separation, R12, the model yieldsthe relative rotation rate !,

! ¼ k1 þ k2

R212

; ð1Þ

Batchelor (1980), where k1 and k2 are theirrespective ‘‘intensities’’, i.e. the integrated rela-tive vorticity over the cyclone’s core. Theirmotions are directed perpendicular to their con-nection line and their respective individualspeeds, V1 and V2, obey the ratio

V2

V1

¼ k1

k2

: ð2Þ

The binary rotation has, therefore, the followingcharacteristics: (a) Cyclonic relative rotationwith (b) a rate proportional linearly to thecombined intensity and (c) inversely to thesquare of the separation distance while (d)the weaker cyclone rotates faster around theintense cyclone.

Several studies of the binary interaction (e.g.Haurwitz, 1951) has shown that for interactingcyclones with finite cores that do not overlap(remote interaction), Eqs. (1) and (2), hold.Therefore, they may be considered as valid forremote binary interactions in general, rather thanonly for two point vortices.

Extensive studies of binary tropical storms(e.g. Hoover, 1961; Brand, 1970; Dong andNeuman, 1983) have confirmed the existence of(a)–(c) in the tropical Pacific, but not in the trop-ical Atlantic. Brand (1970) found cyclonic rota-tion in the Pacific for cyclone pairs withseparation distance smaller than 1300 km. Thedisagreement between theory and observationsin the Atlantic was attributed to the effect ofthe background anticyclonic flow on the binarycyclones there (Dong and Neuman, 1983; Landerand Holland, 1993).

Extensive observational studies have indicatedthe existence of interactions that are far morecomplicated than the simple Fujiwhara effect,including rapid merging in some occasions andescape in others. The tropical binary interactionwas also studied numerically. For example,Chang (1983) and DeMaria and Chen (1984),using a barotropic nondivergent model, showeda high sensitivity of the cyclones’ tracks to theindividual distribution of their tangential veloc-ity. Richie and Holland (1993) and Holland and

Dietachmayer (1993) studied the complexity of‘‘non-Fujiwhara’’ effects through barotropicmodels, accounting for the role of the cyclones’fine structures and the convective cloudinesswithin them. Falkovich et al. (1995) examinedthe role of both beta effect and ocean feedbackupon tropical binary storms. They found the betaeffect to be a source of an asymmetry betweenthe development of the intensities and tracks ofthe individual cyclones that take part in a binaryinteractions, whereas the ocean feedback has adissipating effect on both this asymmetry andthe system’s intensity as a whole. Lester et al.(1998) classified the non-Fujiwara effects foundamong tropical cyclones in the north Atlanticand differentiated between internal factors, thatare associated with an asymmetry between theinteracting cyclones themselves, and externalfactors, that deflect the motion of one or bothcyclones from that implied by the pure Fujiwaraeffect.

Ziv and Alpert 1995 (ZA95, hereafter) studiedthe rotational aspect of 1423 cyclone pairs thatwere found within the 30�–60�N, 0�–60� Edomain for 24 winter months. An example of acyclonic rotation is shown in Fig. 1. ZA95 foundthat when the separation distance between twocyclones is 2000 km or less, they rotate cycloni-cally on the average. They also found that in thesubtropics (15�–30�N) the average rotation isanticyclonic, and attributed this to both theupper-level anticyclonic shear and to the preva-lence of surface anticyclones there. In the mid-latitudes the background shear was found to havea significant enhancing effect only on cyclonepairs with separations smaller than 900 km.Renfrew et al. (1998) found binary rotationwithin pairs of Polar Cyclones with a good agree-ment with theory, especially when they reachtheir secondary (convective) stage.

The observed features of binary systems at themid-latitudes found by ZA95 may be summa-rized as follows (also see Table 1).

* Sense of rotation: The rotation is cyclonic onthe average up to 2000 km separation.

* Intensity=rotation relationship: The rotationrate increases with the sum of the intensities ofthe interacting cyclones.

* Separation=rotation relationship: The rota-tion rate decreases with separation distance

190 B. Ziv and P. Alpert

up to about 1400 km, then the curve reversesits slope between 1400 and 1900 km with apronounced peak near 1900 km (Fig. 2).

* Intensity=rotational speeds ratio within thebinary systems: No significant correlationwas found.

Fig. 1. 1000 hPa geopotential height distribution in dm with 1 dm interval for: (a) 00 UTC 22 Dec. 1986, (b) 12 UTC 22 Dec.1986, (c) 00 UTC 23 Dec. 1986 and (d) 12 UTC 23 Dec. 1986. Cyclones composing the cyclone pair are denoted by A and B(following Ziv and Alpert, 1995)

Table 1. The characteristics of binary interaction according to the point vortex theory against that of the observed mid-latitudebinary cyclones (following Ziv and Alpert, 1995)

Characteristic Findings Comments

Cyclonic rotation up to 2000 km separation observed values agree with theoryRotation=separation relation agreement up to 1400 km unexplained peak near 1900 kmRotation=intensity relation agreement ‘‘cyclone intensity’’ – its geopotential minimumSpeed-intensity relation no significant relation theoretical relation was found at 500 hPa level

Rotation of mid-latitude binary cyclones: a potential vorticity approach 191

The above findings agree with the point vortexmodel, except for two disagreements: one is thereversal of the predicted separation-rotation rela-tion for separations of 1400–1900 km, second isthe lack of correlation between rotation speedsand intensities within the binary systems.

ZA95 attributed these disagreements to thebaroclinicity typifying the mid-latitudes, whichdoes not exist in the 2-D point vortex model.Indeed, the disagreements nearly disappeared ina parallel analysis of the 500 hPa binary systems(ZA95), perhaps due to the more barotropic-likenature of the flow at this level, as compared tothat on the surface.

Haurwitz (1951) stated that ‘‘it seems unlikelythat the theory in its simple form can be appliedgenerally to extratropical cyclones and anticy-clones because the deviations from barotropicstratification are presumably large, so conditionsaloft are quite different from those near the sur-faces.’’ This statement and the findings described

above provided our motivation to extend the tra-ditional approach beyond the 2-D framework inorder to better describe qualitatively the mid-lati-tude binary interactions and to try explaining thedisagreements between their observed features ascompared with both that found in the tropics andthat predicted by the 2-D model. We replace the2-D vorticity by the 3-D counterpart, i.e. bythe Ertel Isentropic Potential Vorticity (PV).The use of PV concepts is also called ‘‘PV think-ing’’ (Hoskins et al., 1985; HMR hereafter). ‘‘PVthinking’’ considers a surface cyclone as a reflec-tion of anomalies either in the upper-PV or in thesurface-� fields, implying that the use of a sur-face pressure chart for studying the binary inter-action may yield misleading results.

The next section outlines the two basic typesof cyclones according to ‘‘PV thinking’’ and dis-cusses their implications to the binary interac-tion. Section 3 demonstrates the hypothesizedfeatures through the analysis of upper-PV andlower-� fields in association with two mid-lati-tude binary episodes. Section 4 describes a dis-tinct group of mid-latitude binary cyclones thatwe have identified and discusses its unique rota-tional features. The last section summarizes theresults and their implications to the observedmid-latitude binary interactions.

2. Binary interaction as interpretedby ‘‘PV thinking’’

The potential vorticity (PV) is given, underhydrostatic conditions, by

PV ¼ g�@�

@p; ð3Þ

(following Rossby, 1940 and Ertel, 1942), wherep is pressure, g is gravity acceleration, � is thevertical component of the absolute vorticity,derived on isentropic surfaces, and � is the poten-tial temperature. PV and potential temperatureare the only Lagrangian invariants of a 3-Dadiabatic and frictionless flow (Egger, 1989).Large and synoptic-scale atmospheric systemscan be assumed adiabatic and frictionless(Holton, 1992). For ‘‘balanced dynamicalphenomena . . . the dynamical evolution can bedescribed solely in terms of the PV (and surfacepotential temperature) distributions, their in-duced wind fields and their advective rates of

Fig. 2. Average rotation factor as a function of separationfor the period 1982–1988, months December–March forthe domain 30–60� N=0–60� E of all cyclone pairs (solid),‘isolated’ pairs (dashed) and for ‘isolated’ and ‘free’ pairs(dotted). The semi-dashed line shows the 2-D theoreticalrelationship. ‘Isolated’ pairs are pairs around which noother cyclone was found within 1000 km radius, and ‘free’pairs are pairs that any of its member was not located withina cyclogenetic area (following Ziv and Alpert, 1995)

192 B. Ziv and P. Alpert

change’’ (HMR). The ‘‘Invertibility Principle’’states that ‘‘given the PV distribution, one candeduce the wind, pressure and temperature field’’(HMR).

Inversions of PV and surface potential tem-perature fields show that a surface cyclonereflects a positive anomaly in the upper-levelPV or in the surface potential temperature(Thorpe, 1985; HMR). Moreover, each of sucha positive anomaly induces cyclonic flow overits surroundings. Hence, one may expect thedetailed interaction among neighboring cyclonesto depend on the structures of the 3-D flow pat-terns induced by their ‘source anomalies’. In thefollowing discussion, a cyclone associated withan upper-level PV anomaly will be referred to asa ‘‘cold cyclone’’ and that associated with asurface thermal anomaly as a ‘‘warm cyclone’’though in real situations a cyclone may reflect acombination of both (e.g. a baroclinic cyclone,HMR). Figure 3 shows cross-sections throughthese two cyclone types following Thorpe(1985) as appears in HMR. They were derivedby inverting the PV field for uniform conditions,with a positive circular symmetric near-tropo-

pause PV anomaly (a) and a surface thermalanomaly (b), of 1667 km radius. Each type ofanomaly induces a cyclonic tangential flow,attaining its maximum within its periphery andextending out of its boundaries, thus enabling aremote interaction with neighboring anomalies.The induced flow patterns of the two types differsubstantially in their vertical distribution. Theinduced flow of the upper-PV anomaly maxi-mizes at the tropopause and falls to about 60%of its maximum speed near the surface, whilethat induced by a surface thermal anomaly max-imizes at the surface, but falls sharply withheight, reaching only about 20% of its maximumvalue at the tropopause level.

Of course, the vertical penetration of theinduced flow depends on the horizontal scale ofthe pertinent anomaly, but in the context of bi-nary interactions it can be concluded that upper-PV anomalies tend to drive a surface-thermalanomaly, and their associated warm cyclone,faster than in the opposite direction, i.e. thatlower-thermal anomalies would drive upper-PVanomalies. Hence, the upper anomalies, whenthey exist, dominate the binary interactions.

The nature of the inversion operator (HMR)implies that both the speed of the induced flowand the intensity of the surface cyclone would beproportional to the intensity of its source anom-aly. Therefore, the advective power of each ofthe interacting surface cyclones may, to a firstapproximation, be proportional to its intensity,as implied by the 2-D approach. This may be arather realistic assumption for a system consist-ing of two warm cyclones, since the sourceanomalies and their associated cyclones are atthe same level. But, for cold cyclones this isnot generally true, since the intensity of a surfacecyclone is also affected by the lower-level ther-mal field. Moreover, upper-troposphere positivePV anomalies often tend to coincide with lower-cold anomalies (HMR; Alpert, 1984) whichreduce the intensity of the surface cyclones asso-ciated with the former. Hence, in binary systemsassociated with at least one upper-PV anomaly,the correlation between the ratios of the rotationspeeds and the intensities of the surface cyclonesis expected to be poor. The above can be general-ized by saying that when two upper-PV anoma-lies are involved in a binary interaction, therotation rate of their associated cyclones would

Fig. 3. Cross-sections through circular symmetric circula-tions induced by isolated positive anomalies (whose loca-tion is shown stippled) a in near-tropopause PV and bsurface potential temperature anomalies produced byHMR. Thin lines represent both isentropes and tangentialvelocity. Bold lines indicate the tropopause

Rotation of mid-latitude binary cyclones: a potential vorticity approach 193

obey Eqs. (1) and (2), but with respect to theintensity of the upper-anomalies rather than withthat of the surface cyclones.

The PV approach shifts our attention from thesurface cyclones toward the source anomalies asthe interacting objects, so the concept ‘‘binarysystem’’ would be preferred over ‘‘binarycyclone’’ for the mid-latitudes. The main impli-cation of this approach is that whenever upper-PV anomalies exist in a binary system, theydominate the rotation of the system.

3. Analysis of the observedbinary systems

The observational study is based on ten cases ofbinary interaction that lasted for at least 36 hours,in which the binary system completed rotation of70� or more. For each case a time series of sur-face pressure charts presents the rotation process,the lower-level thermal field and the upper-levelPV-field, show the structure and rotation of therespective source anomalies. Two cases are pre-sented below. They represent two different typesof binary systems: in the first, one of the twointeracting cyclones is cold while the other isbaroclinic, i.e. associated with the two types ofsource anomalies at different locations. In thesecond case, one of the interacting cyclones iscold and the other – warm. No case of pure warmbinary system was found north of 30�N.

The analysis makes use of ECMWF initializeddata (Bengtsson et al., 1982; Haseler andSakellarides, 1986; Hollingsworth et al., 1986)containing three wind components, geopotentialheight and temperature on seven mandatory pres-sure levels for the area 0�–60�N, 0�–60� E. Foreach case, three fields are shown: First, the1000 hPa geopotential height, representing thesurface pressure; second the 850 hPa potentialtemperature, representing the surface thermalfield; and third, the PV on the isentropic levelwhich represents, more or less, the tropopausefor the relevant case. The 850 hPa surface waschosen for representing the ‘‘surface’’ thermalfield in order to avoid the small scale featuresand the diurnal effects within the planetaryboundary layer (HMR). The isentropic PV fieldwas extracted from the isobaric data, assuminglinear dependence of temperature on the log-pressure between successive pressure levels,

which was found by Ziv and Alpert (1994) tobe superior to other interpolation methods.

The cases presented below were extracted bythe analysis system developed by Neeman andAlpert (1990). The surface interacting cyclones,i.e. the 1000 hPa geopotential minima, aredenoted ‘‘A’’ and ‘‘B’’, and the increments oftheir motions within the ensuing 36 hours, at12 h intervals, are depicted. Several approachesfor identifying an ‘anomaly’ may be used. Thissubject is discussed in Appendix A. Here, a localmaximum or a pronounced ‘‘tongue’’ in thethermal=PV field is considered as an anomaly.A notation ‘‘A’’ or ‘‘B’’ refers to the sourceanomaly of the respective surface cyclone. Foreach case the relative rotational speeds of theinteracting surface cyclones are compared withtheir individual intensities, then with that of theirassociated PV anomalies. Due to the illustrativenature of this part of the study the intensities areevaluated qualitatively.

3.1 Case 1: 3–4 January 1985,cold-baroclinic binary interaction

Figure 4 shows two cyclones, denoted A and B,located initially, in 00 UTC 3 January 1985, in

Fig. 4. 1000 hPa geopotential height distribution in dm,with 1 dm interval, for 00 UTC 3 Jan. 1985. Cyclonescomposing the pair are denoted by A and B. The locationof the source PV anomaly of cyclone A and the source PVand � anomalies of cyclone B are denoted APV, BPV and B�,respectively. The arrows represent their movements for 3time increments of 12 h

194 B. Ziv and P. Alpert

Lithuania (57�N, 22� E) and in the Balkans(42�N, 23� E), respectively, subjected to a cy-clonic rotation. The imaginary line that joins themrotated about 120� during 36 hours with only aslight decrease in length (R), about 1500 km.Cyclone B, however, moved much faster thancyclone A, though it was the deeper one, asreflected from the minimum in geopotentialheight. It may be argued that since both cycloneswere steered by the mid-latitude upper-levelwesterlies, the relative westward progression ofcyclone A was reduced and the eastward progres-sion of cyclone B was enhanced. However, themeridional displacement of cyclone A was alsosmaller, about 4� latitudes, compared to 15� forcyclone B, suggesting that the latter indeed had alarger rotation speed. Figure 5 shows the 320 Kisentropic PV field for 00 UTC 4 January 1985.Two positive PV-anomalies coincide more or lesswith the cyclones’ locations. Anomaly A appearsin Fig. 5 as a pronounced tongue that extendedfrom the north and disintegrated 12 hours later,and B was a local maximum. They rotated in thesame fashion as the surface cyclones. The centralPV value of anomaly A was larger than that ofanomaly B by over 0.5 PVU, implying thatanomaly A was the more intense. Figure 6 shows

the 850 hPa � field for 00 UTC 4 January 1985.No thermal anomaly is found in the vicinity ofcyclone A, but a maximum was found north ofthe Black Sea (48�N, 42� E), about 6� to thesouth east of the cyclone B. This anomaly,denoted B� in Fig. 4, progressed toward thenorth–east, maintaining its location relative tothe cyclone. The thermal anomaly B� mayexplain why the location of cyclone B did notcoincide with the upper PV anomaly B, but tothe east, between the two source anomalies.The combined contributions of both anomaliesexplain also why cyclone B was deeper thancyclone A by 40 m (Fig. 4). However, the rota-tion speed of cyclone B does not seem to berelated to its intensity relative to cyclone A, butrather to that of its source upper anomaly.

3.2 Case 2: 10–11 January 1984,cold-warm binary interaction

Figure 7 shows a cyclone pair, denoted A and B,located over south Italy (41�N, 15� E) and Tunis(36�N, 8� E), respectively, undergoing a cyclonicrotation. Cyclone A remained nearly stationary(except for a slight southward movement) whilecyclone B revolved around it counterclockwiseby about 80� within 36 hours. This seems tocontradict the observed similarity between the

Fig. 5. Isentropic 320 K PV distribution in 1 PVU(PVU¼ 10� 6 m2 s�1 K kg�1), with 0.4 PVU interval, for00 UTC 4 Jan. 1985. Areas with values exceeding 6 PVUare shaded. The positive anomalies corresponding to cy-clones A, B are denoted, respectively

Fig. 6. 850 hPa potential temperature distribution for thesame time as in Fig. 5. The anomaly associated with cy-clone B is denoted

Rotation of mid-latitude binary cyclones: a potential vorticity approach 195

intensities of the two cyclones for the major partof the period, i.e., their geopotential heights dif-fered by only 20 m. The upper PV and the lowerthermal fields, however (Figs. 8 and 9 respec-tively), indicate that cyclone A coincided with

a PV maximum centered over central Italy,whereas cyclone B was associated with a thermalanomaly extending from northern Africa to themiddle Mediterranean and later moved eastward.The asymmetry between the types of their sourceanomalies is suggested here to explain whycyclone B dominated the rotation.

These two cases are in line with the hypothe-sized features of the binary rotation described inSec. 2. One feature is that in an interaction inwhich two upper-PV anomalies are involved,the cyclone that is associated with the moreintense anomaly moves slower. The second isthat in a warm-cold interaction, the warmcyclone revolves around the cold one and doesnot follow the ratio of the intensities of the sur-face cyclones, i.e. Eq. (2). A further analysis ofthe above cases, as well as other eight cases (notshown), has yielded a refined description of aunique type of mid-latitude binary system, asdescribed in the following section.

4. The contact binary system (CBS)

So far the binary systems were considered asremotely interacting. This implies that each ofthe two individual anomalies involved in a binaryinteraction is assumed to be small as comparedwith the separation distance between them. Thedata analysis of the mid-latitude binary systems,

Fig. 8. Isentropic 330 K PV distribution in 1 PVU, with 1PVU interval, for 00 UTC 11 Jan. 1984. Areas with valuesexceeding 3 PVU are shaded. The positive anomaly corre-sponding to cyclone A is denoted

Fig. 9. 850 hPa potential temperature distribution for thesame time as in Fig. 8. The anomaly associated with cy-clone B is denoted

Fig. 7. 1000 hPa geopotential height distribution in dm,with 1 dm interval, for 00 UTC 10 Jan. 1984. Cyclonescomposing the pair are denoted by A and B. The locationsof their source anomalies are denoted APV and B�. Thearrows represent their movements for 3 time incrementsof 12 h

196 B. Ziv and P. Alpert

however, have indicated that in quite a number ofcases one of the interacting anomalies is rela-tively large, so that the other is located at itsperiphery. This has led us to propose the term‘‘contact binary interaction’’ for such a case,and the name ‘‘Contact Binary System’’ (CBS)for the pertinent system. The CBSs were found tohave common unique features. This section out-lines these features. First the unique structure ofa CBS is described, then its rotational character-istics are derived. The last subsection discussesthe impact of their existence on the observedabnormal increase in the rotation rate withseparation distance of mid-latitude binary sys-tems in the 1400–1900 km range found by ZA95.

4.1 Structure

The main features of a CBS are shown schemat-ically in Fig. 10. One of the cyclones, L1, is asso-ciated with a major upper PV anomaly, oftenassociated with closed PV-contours. The othercyclone, L2, is associated with a thermal anom-aly, sometimes combined with an upper-PV sec-ondary anomaly, at the periphery of the primaryone, to its south–east (e.g. case 2 above). TheCBSs may explain the higher observed frequencyof binary cyclones oriented in a northwest–southeast direction (Fig. 11). Cyclone L1 willbe referred to as the primary cyclone, andcyclone L2 as the secondary one.

In cases where the secondary cyclone reflectscombined contributions of two anomalies,

cyclone L2 may be more pronounced thancyclone L1 at the surface. But, since it is as-sociated with the weaker upper-PV anomaly,cyclone L2 rotates faster than L1. In this way,the CBSs contribute to the discrepancy in theintensity=rotation relationship for the surfacebinary systems as found by ZA95. Moreover,the typical location of the primary cyclone L1

to the north of the secondary cyclone L2

(Fig. 10) implies that L1 is expected to move tothe west, but this movement is suppressed by thesteering effect of the upper-level westerlies,resulting in its stationary appearance (Sec. 3).

The tendency for the secondary anomaly (L2)to be formed and to move along the periphery ofthe primary cyclone (L1) may be understood, fol-lowing HMR. The primary L1 positive PV anom-aly, resulting from an equatorward displacementof a polar air mass, possesses low temperaturesand high PV. Consequently, the boundaries of the

Fig. 10. A schematic picture of a typical mid-latitudeContact Binary System (CBS). Solid lines are upper-levelPV isolines and dotted-thin lines are surface isotherms. Thesurface cyclones, denoted L1 and L2, are represented bydotted circles and their velocities by thick arrows

Fig. 11. Polar diagram showing the distribution of theorientations of mid-latitude binary cyclones that wereobserved (ZA95). This is based on the subset of 327 cyclonepairs, each of which had a separation less than 2100 km, freefrom any additional cyclone within 1000 km distance andlocated out of cyclogenetic areas (ZA95). Distance fromthe center is proportional to the number of occurrences. Eachsector is of 45� and is represented by its central point. The270�–315� (90�–135�) sector, for instance, is the most fre-quent with 15% of the occurrences

Rotation of mid-latitude binary cyclones: a potential vorticity approach 197

L1 anomaly are highly baroclinic. This baroclini-city maximizes along the southern boundaries, sothis sector becomes favorable for the formationand maintenance of disturbances, such as the L2-type cyclones. The latter are, indeed, frequentlyfound at the frontal zone, at the south–east sectorof a Mid-latitude ‘parent’ cyclone. The SharavCyclones that tend to form in the spring seasonalong the North African coast (Egger et al., 1994;Alpert and Ziv, 1989) are examples for such asystem.

4.2 Rotation=separation relationship

Next, a relationship between the rotation ratesof the CBSs and their separations is derived.The derivation is based on the followingassumptions:

* The primary anomaly is assumed to be theresult of an equatorward extension of anupper-level trough that turns into a cutoffcyclone (e.g. Palmen and Newton, 1969).

* The secondary anomaly is significantly smal-ler, both in size and in relative vorticity, com-pared to the primary one so

* The rotation of the CBS as a whole is deter-mined by its motion around the primary(Fig. 10), and

* The velocity of the secondary cyclone isapproximated by the tangential wind alongthe periphery of the primary anomaly.

* The wind and relative vorticity within the pri-mary anomaly have both circular symmetry.

Consider a polar air mass with zero relativevorticity that is centered at a polar latitude �i.The air mass possesses an absolute vorticityf ð�iÞ, larger than those characterizing themid-latitudes, and is displaced equatorward toa mid-latitude �f (Fig. 12). Conservation ofabsolute vorticity implies that when migrating,the air-mass would acquire positive relativevorticity ð�Þ with a central value of �max, whichis opposite and equal in magnitude to therespective change in the Coriolis parameter,�f , i.e.

�max ¼ ��f ¼ f ð�iÞ � f ð�f Þ: ð4ÞBased on the beta-plain approximation,

�f ¼ � ��y; ð5Þ

where �y is the meridional displacement of theair mass and � � @f

@yffi 2� cos ð’0Þ

a. Substituting 45�

for ’f and assuming that �y is equal to the radiusRm of the air mass (Fig. 12) yield

�max ffiffiffiffi

2p

�

aRm: ð6Þ

If the meridional gradient in vertical stratifica-tion is not accounted for, Eq. (3) implies that theamplitude of the PV anomaly would be equal to�max.

Actually, the air mass would adjust itself sothat the PV would be partitioned betweenvertical stratification and absolute vorticityaccording to its horizontal scale. Under quasi-geostrophic conditions the vorticity anomalycan be deduced from an anomaly in the streamfunction, , given by

¼ 0e�r=Re�f ðz�z0Þ=NH; ð7Þwhere r and z are horizontal and vertical coor-dinates, R, H are the horizontal and verticalscales, respectively, N is the Brunt-Vaisala fre-quency and z0 is the height of the anomaly.Following HMR, the vorticity anomaly, �, andvertical stratification anomaly, s, are obtainedfrom the second derivatives in r and z, respec-tively, i.e.

� ¼ 1

R2 ; s ¼ f 2

N2H2 : ð8Þ

The approximate ratio between the vorticity atthe anomaly center, �0, and the maximum PVanomaly, �max, is, therefore,

�0

�max

ffi 1

1þ f 2R2

N2H2

ð9Þ

Fig. 12. Scheme of a circular upper-level PV anomaly(shaded) with radius R. The center originated from thelatitude �i and has moved to the latitude �f

198 B. Ziv and P. Alpert

The same relation is obtained when a Gausiandistribution of is applied in (7). Inserting (9)in (6) yields

�0 ffiffiffiffi

2p

�

a

Rm

1þ f 2R2m

N2H2

!

ð10Þ

The tangential speed along the periphery, V , maybe obtained from the circulation, C, which isgiven, following the Gauss theorem, by integrat-ing the vorticity over the anomaly’s core, i.e.

C ¼ 2�

ðRm

0

�r dr ¼ 2�

ðRm

0

�0 F

�

r

Rm

�

r dr

¼ 2��0�R2m; ð11Þ

where Fðr=RmÞ is a dimensionless circularlysymmetric vorticity distribution and

� �ðRm

0

F

�

r

Rm

�

r dr; ð12Þ

The tangential wind speed is, then,

V ¼ C

2�Rm

¼ �0�Rm; ð13Þ

and the rotation rate ! is

! ffi � V

Rm

¼ ��0

¼ �ffiffiffi

2p

�

a

Rm

1þ f 2R2m

N2H2

!

: ð14Þ

Assuming that the vorticity distribution withinthe anomaly, and hence �, are independent on theanomaly size, the dependence of �0 on Rm forvarious CBSs, determines the !-R relationship.The rotation attains its maximum at

Rm ¼NH

f: ð15Þ

A substitution of 10�2, 104 and 10�4 for N, Hand f , respectively, in (15) indicates that themaximum rotation rate would be found arounda 1000–1500 km separation distance. The exis-tence of CBSs explains the observed increasein the rotation-separation relationship foundbetween 1400 and 1900 km separation (Fig. 13).

4.3 Quantitative estimateof the rotation rate

In order to evaluate the expected rotation ratesfor CBSs the dimensionless vorticity distribution

� and the scaling factors, H and N, must be spec-ified. It is worth noting that � has a key role inthe detailed binary interaction in the tropics(Chang, 1983; DeMaria and Chen, 1984). Twovortex types are most common in the literature.One is the ‘‘patch vortex’’, or the ‘‘Rankine vor-tex’’, in which the relative vorticity is uniform allover its core. This type was adopted by Haurwitz(1951) for the tropical storms, but seems to beless appropriate for mid-latitude cyclones, wherea pronounced decrease in vorticity with distancefrom the vortex center is found (not shown).Another type is the ‘‘cosine vortex’’, for whichFðr=RmÞ ¼ 0:5½ cos ð�r=RmÞ þ 1�. This type wasapplied by Thorpe (1985) for temperature depar-tures from normal within the PV and � anomalies.The values of � are 0.5 for the Patch vortex and0.15 for the Cosine vortex. The actual PV distri-butions within positive anomalies (see Figs. 5, 8)seem to have distributions which are between theuniform and cosine distributions, so � ¼ 0:3 wasadopted. The values applied for H and N are8�103 m and 10�2 s�1, respectively.

Figure 13 shows 3 curves of rotation rate as afunction of separation, i.e., the observed forisolated-free pairs (Fig. 2), the theoretical point

Fig. 13. Variation of average rotation factor for cyclonepairs as a function of separation for isolated-free pairs(Fig. 2). Dashed line represents the theoretical point vortexrelation (Eq. (1)) and the semi-dashed line – the relation forthe CBS proposed here (Eq. (14)) with � ¼ 0:3

Rotation of mid-latitude binary cyclones: a potential vorticity approach 199

vortex relationship (Eq. (1)) and the relation corre-sponding to the CBS proposed here (Eq. (14)). Theobserved curve follows, more or less, the pointvortex curve for small separations up to 1400 km,then changes its slope abruptly and approachesthe values corresponding to that of the CBS near1900 km separation and finally drops sharply andintersects the zero line at about 2100 km. Thesharp increase in the observed R-! relation canbe attributed to an increase in the proportion ofCBSs to the total binary systems, suggestingthat larger upper-level cutoff cyclones are morefavorable for formation of secondary distur-bances, and hence for CBSs. Moreover, the1400 km distance seems to be a lower limitfor CBS formation. The sharp decrease of theR-! relation for separations around 2000 kmmay be interpreted as the combined result ofan upper limit of the upper-level cutoff cy-clones’ radius and the approach to the influencerange of binary interaction, being 2000 km(ZA95).

5. Summary and discussion

This study is the first attempt to describe thebinary interaction from a PV viewpoint. Thisapproach is adopted here for the mid-latitudebinary cyclones, and is used to study their rota-tional features. ‘‘PV thinking’’ refers to surfacecyclones as the projections of upper-PV or lower-� positive anomalies or some combination ofboth. The traditional 2-D approach is extendedhere to the 3-D domain, enabling the inclusionof baroclinicity, which characterizes the mid-latitudes. Our study differs from the traditionalpoint vortex approach in two aspects:

* The PV and thermal anomalies are consideredas the interacting objects rather than thesurface cyclones alone.

* The contact interaction is introduced and,along with the remote interaction, is shownto provide a more comprehensive picture ofour earlier findings.

The study has two objectives: one is to betterdescribe the structure and behavior of the mid-latitude binary systems. Second, is to explain thedisagreements between the observed rotationalfeatures of the mid-latitude binary cyclones(ZA95) and the predictions of the point vortex

model, which agrees with the observed rotationof tropical binary cyclones.

The PV concepts, when applied to binary sys-tems, lead to the following conclusions:

* Whenever significant upper-PV anomalies areinvolved, they dominate the rotation process.

* Consequently, the individual rotation speedof each surface cyclone is proportional to theintensity of the source upper-anomaly of theother cyclone rather than with its own intensity.

The latter explains the lack of correlationbetween the intensities of the surface cyclonesand their rotation speeds within mid-latitude bi-nary systems, found by ZA95, as predicted by thepoint vortex model. These conclusions are illus-trated by case studies, two examples of which areshown in Sec. 3 above.

A further examination of data has indicated theexistence of a distinct type of mid-latitude binarysystem. This type is entitled here ‘‘Contact BinarySystem’’ (CBS). The CBS is composed of onecyclone that is associated with a major upper-PVanomaly and another, located at its southeast pe-riphery, normally associated with a lower-thermalanomaly, sometimes combined with a secondaryupper-PV disturbance. Analytical considerationsindicate that for CBSs the rotation rate has aunique dependence on separation distance, witha maximum near 1300 km separation.

Data analysis, when interpreted through PVthinking, yields the following conclusions:

* A mid-latitude binary system is associatedwith at least one upper-PV anomaly.

* The dominance of the upper-PV anomaliesover the rotation process implies that the in-dividual rotational speeds of the interactingcyclones are better correlated with the relativeintensities of the upper-level anomalies thanwith those of the surface cyclones. Thishypothesis agrees with the analysis of binaryinteractions at the 500 hPa level, ZA95.

* The population of mid-latitude binary systemsis partitioned between remotely interactingand contact interacting systems (CBSs).

* CBSs are suggested here as the contributors tothe observed increase in separation=rotationrelationship for 1400 to 1900 km separationdistances, ZA95, and a theoretical basis forthis increase is derived.

200 B. Ziv and P. Alpert

‘‘PV thinking’’ probably explains also the dif-ference between the maximum distance wherebinary interaction dominates in the mid-latitudes(2000 km, ZA95) and in the tropics (1300 km;Brand, 1970). Since the binary interaction inthe mid-latitudes is dominated by cold cycloneswhile that in the tropics – by warm cyclones, thisdifference may represent a respective differencein their radii of influence. The wind patterns thatwere derived for the two cyclone types (HMR,Fig. 3) differ, indeed, in their horizontal distribu-tion. The surface wind at the lateral boundariesof the domain is about 20% of its maximumvalue for the former, but only to about 10% forthe latter, though the horizontal radii of the twosource anomalies are the same.

An interaction between more than two neigh-boring cyclones may sometimes occur, withincreased complexity in the motion of individualcyclones (Aref, 1983). A relatively simple exam-ple is the interaction between the intense polarvortex and the mid-latitude cyclones. This vor-tex, with its large vertical scale and horizontaldimensions, and its induced mid-latitude wester-lies, drives the relatively weaker mid-latitudeanomalies, and their associated surface cyclones,to revolve around it. At the same time, mutualinteractions among these relatively weak anoma-lies are superposed upon the major interaction.

The differentiation between binary rotationand the steering by the ambient flow is not elab-orated here. If both anomalies were advected bythe same flow, this effect could be eliminated bysimply evaluating the rotation of the vector con-necting the two rotating cyclones, as is donehere. But, if each anomaly is steered by a differ-ent ambient flow (as is the case for surface-uppersystem) the contribution of the differentialadvection may complicate the situation. Assum-ing that the ambient flow is zonal, this effectcan affect the relative motions of the rotatingcyclones when the cyclones are located at differ-ent latitudes. This is relevant to case 2 (Fig. 7),when the upper-level flow is expected to driveanomaly (A) eastward and the surface flow isexpected to drive anomaly (B) westward, imply-ing a negative contribution to the relative rotationof the cyclone pair. The significant positive rota-tion observed suggests that the effect of the meanflow is secondary with respect to that of themutual interaction (ZA95).

The PV approach, when used for describingmid-latitude binary cyclones qualitatively, isfound here capable of describing the mid-latitudebinary interactions and to explain the discrepan-cies between the barotropic theory and observedcharacteristics. This work can serve as a pre-liminary conceptual basis for future quantitativestudy, based on inversion of PV fields.

Appendix A: identification of anomalies

The concept of ‘anomaly’ refers to a local or regional devia-tion of a field from a reference value. The amplitudes andgeometrical shapes of the anomalies depend on the choice ofthe reference state. Some examples follows.

(a) A uniform reference state. HMR use anomalies (Fig. 3)relative to a uniform reference state. Therefore, the PVfield at a certain isentropic level is displayed after thesubtraction of a constant value. This approach preservesthe original distribution regardless of the chosen refer-ence state’s value.

(b) A zonal reference state. A PV or � anomaly may also bedefined relative to a nonuniform, e.g. zonal, referencestate. An anomaly may thus change its amplitude as itmigrates meridionally, since only its absolute value is tobe conserved. The choice of a zonal reference state maybe made in several ways. One is to multiply the localCoriolis parameter by the zonal average stability param-eter, ignoring the large-scale circulations (HMR). Othermay be obtained by averaging the PV over the latitudinalbelt or by multiplication of the average absolute vorticity

Fig. 14. Isentropic 360 K PV-distribution for 00 UTC Jan. 31985 in PVU with an interval of 1 PVU (solid). The areaswhere the Laplacian of the PV is negative, indicating posi-tively anomalous PV, are shaded. Three PV maxima are foundnorth of 30� N, whereas the Laplacian has at least 10 maximain addition to those which coincide with the PV maxima

Rotation of mid-latitude binary cyclones: a potential vorticity approach 201

by the average stability parameter. These approaches arenot identical. Due to the inherent ambiguity of theseapproaches, they were not adopted.

(c) Mapping the Laplacian of the PV. Mapping the Lapla-cian of any field enhances local deviations in the relevantvariable, i.e., anomalies. However, this operator is verysensitive to small-scale features, what makes it hard tovisualize the synoptic scale systems, see e.g. Fig. 14.HMR stress that PV charts magnify the synoptic-scalesystems, because of the laplacian-like nature of the rela-tive vorticity field with respect to the geopotential heightfield.

For the above argument a local maximum or a pro-nounced ‘‘tongue’’ of positive PV or potential tempera-ture is referred to as an anomaly in this study.

Acknowledgments

We thank the German Israeli Foundation (GIF) I-138-120.8=89 for supporting this research. Thanks to ECMWFfor the data, to Prof. B. Hoskins and Prof. J. Egger, whoreviewed this paper, and to Prof. M. Mclntyre, Prof.A. Thorpe, Dr. E. Heifetz and Dr. A. Tafferner for valuablediscussions. Thanks to support from the EU project DETECTand to the GLOWA-JR project funded by BMBF and MOS.

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Authors’ addresses: Baruch Ziv (baruchz@openu.ac.il),The Open University of Israel, 16 Klousner St., Tel-Aviv61392, Israel; Pinhas Alpert (e-mail: pinhas@cyclone.tau.ac.il), Department of Geophysics and Planetary Sciences,Raymond and Beverly Sackler Faculty of Exact Sciences,Tel Aviv University, Tel Aviv 69978, Israel.

202 B. Ziv and P. Alpert: Rotation of mid-latitude binary cyclones: a potential vorticity approach