DECEMBER 1996 DESROCHERS AND HARRIS 2191Interpretation of Mesocyclone Vorticity and Divergence Structure from Single-Doppler Radar PAUL R. DESROCHERSPhillips Laboratory/GPAB, Hanscom Air Force Base, Massachusetts F. IAN HARRISHughes ST)( Corporation, Hanscom Air Force Base, Massachusetts(Manuscript received I 1 December 1995, in final form 30 May 1996)ABSTRACT A kinematic mesocyclone model is developed to better approximate mesocyclone flows observed by singleDoppler radar. The model is described by two general flow regimes: an inner core region where velocity variesdirectly with radius from the center of the flow and an outer flow region where velocity varies inversely withradius. The new model differs from the traditional circular mesocyclone model in that the shape of the innerflow is described by an ellipse of specified eccentricity, and the vorticity and divergence structures of the innerflow region are nonuniform and described by simple functions. The effects of flow shape, vorticity and divergencestructures, radar viewing angle, and radar resolution on the flow appearance and data interpretation are examined. One traditional measure of mesocyclone intensity is the shear measured between the relative peaks of incomingand outgoing Doppler velocity. In noncircular flows or flows where the vorticity structure is not uniform, shearis found to be an unreliable measure of mesocyclone intensity. A correction for shear is possible if the flowshape, internal structure, and orientation to the radar are known. Techniques to assess these characteristics fromsingle-Doppler data are presented. The elliptical mesocyclone model is compared with observations of the 20 May 1977 Del City, Oklahoma,mesocyclone from two Doppler radars. From characteristics of the flow estimated from single-Doppler data, asimulation of the mesocyclone is produced that closely approximates the observed single-Doppler fields. Theassociated model fields of vorticity and divergence are comparable in structure and magnitude to the fieldsdetermined from dual-Doppler analysis.1. Introduction Mesocyclones are organized rotations associatedwith persistent and intense convective updrafts of severe thunderstorms. These features are often seen as adistinct signature in Doppler radar data to ranges of 200km or more. Because of the strong association betweenmesocyclones and severe weather such as tornadoesand hail (Burgess 1976), their detection and assessment can play an important role in the severe weatherwarning process. Doppler radar detects only the component of flowdirected along the radar beam. Other flow componentsmust be inferred. To a radar scanning in azimuth (Fig.1 ), a rotational flow, such as a mesocyclone, that islarger in dimension than the radar beamwidth will appear as a couplet of diametrically opposed Doppler velocities. This couplet can be interpreted through the ap Corresponding author address: Paul R. Desrochers, PL/GPAB, 29Randolph Road, Hanscom AFB, MA 01731-3010.E-mail: paul @ nomasta.plh.afimilplication of a kinematic model. As a result of a suggestion by Donaldson (1970), it has become commonpractice to approximate the mesocyclone as a circularflow with what is now recognized as a Rankine combined vortex velocity distribution (Doviak and Zrni61984). Under these conditions, vorticity within the mesocyclone core is twice the magnitude of azimuthalshear. Note, however, that not all velocity couplets areassociated with mesocyclones. Donaldson (1970) proposed minimum criteria to identify those coupletswherein rotation was likely: azimuthal shear betweenthe velocity peaks of 0.005 s-~, persistence for half theestimated rotation period, and vertical extent equal tothe velocity couplet diameter. Burgess (1976) reportsthat over 90% of storms that meet these criteria formesocyclone identification produce tornadoes, largehail, or severe straight-line winds. Donaldson's criteriawere further verified during the Joint Doppler Operational Program (JDOP Staff 1979) and are now usedextensively in operations and research. The simplicityof the model and associated criteria paved the way forautomated detection of these features (e.g., Zrni6 et al.1985; Desrochers and Donaldson 1992).2192 JOURNAL OF APPLIED METEOROLOGY VOLUME35D/~=0//~/ /MY'/(m/s) -30 ~ '~ '" / / D [VDopD FIG. 1. Flow field characteristics for the circular mesocyclonemodel. Top frames show horizontal velocity vectors for pm'e rotation(D/~ = 0) and flow where divergence equals vorticity (D/,~ = 1) andwhere R = 2,5 km and Vrot = 25 m s-L The large circle indicates theboundary between inner and outer flow regimes. Simulated Dopplervelocity fields Vvop are for a radar located at the bottom, of figure(middle frames). Doppler velocity contours are every 5 m s-~ withpositive values shaded. Radar resolution is not considered. Profilesof V~op along AB and CD are shown in frames above. Fields of vertical vorticity ~ and horizontal divergence D are shown below withpositive values shaded. Vorticity is 0.02 s-~ within inner flow region. While many tornadoes form without mesocycloneaccompaniment (e.g., Burgess and Donaldson 1979;Wilson 1986; Wakimoto and Wilson 1989), mesocyclonic storms are most notable for the generation oflong-lived tornadoes of F2 intensity or greater. However, not all mesocyclonic storms produce tornadoes,Burgess and Lemon (1990) show that about 30% ofstorms that meet the Donaldson (1970) criteria for mesocyclone classification produce tornadoes. While thereis strong interest in distinguishing between tornadic andnontornadic mesocyclones, limited success has beenachieved. Burgess (1976) found that violent [ >~F4 onthe Fujita (1981) scale] tornado-producing mesocyclones tend to be smaller and have a greater rotationalvelocity than their nonviolent counterparts, but couldnot distinguish between nonviolent tornadic and nontornadic mesocyclones. Donaldson and Desrochers(1990) developed a parameter called ERKE to estimatemesocyclone development at low levels of the stormwith single-Doppler data. This technique proved to beeffective for those storms producing violent tornadoes,but skill drops markedly for lesser tornado intensities.To improve the effectiveness of using mesocyclonestrength to discriminate tornado intensity it appears thata more sophisticated mesocyclone model is required. While considerable practical success has beenachieved through the interpretation of mesocyclones asa circular flow, this approximation is inconsistent withthe many variations in the radial velocity fields of mesocyclones observed by radar. For example, multipleDoppler analyses indicate that mesocyclones can beelongated and have variable vorticity structures (e.g.,Ray 1976; Brandes 1984). Other observations indicatethat the vorticity structure of the mesocyclone undergoes tremendous evolutionary changes prior to tornadoformation (e.g., Brandes 1984). Similar mesocycloneevolution has been observed in results from numericalmodels (e.g., Klemp and Rotunno 1983 ). These studiessuggest the need for greater sophistication in singleDoppler mesocyclone interpretation models. While a circular assumption represents a first-orderapproximation to the flow, a more detailed assessmentof mesocyclone structure from single-Doppler data canbe achieved only through the application of a more general model. In this paper, a new, more robust, mesocyclone model is developed that provides better agreement with observations. In section 2, characteristics of circular 'flows are examined in detail. Interpretation deficiencies of the circular assumption, as they relate to the observations ofthe 1977 Del City, Oklahoma, tornadic mesocyclonesampled at close range by two Doppler radars, are discussed in section'3. In section 4 the new model is presented. The effect of flow shape and the distribution ofvorticity and divergence on mesocyclone appearanceto single-Doppler radar are examined. Finally, in section 5, simulations of the Del City mesocyclone arepresented,2. Circular mesocyclone model A detailed discussion on the interpretation of circularmesocyclone flows via single-Doppler data is providedDECEMBER 1996 DESROCHERS AND HARRIS 2193in Brown and Wood ( 1991 ). The present investigationis concerned with more complex flows. In particular,the impact that flow shape and internal structure mighthave on single-Doppler observations is examined. Theinterpretation of vorticity and divergence structures iscentral to the discussions that follow. The circular model describes an axisymmetric flowcharacterized by two flow regimes: an inner core region where velocity varies directly with radius (r) andan outer potentialflow region where velocity varies inversely to radius. Within the core region, a field of purevorticity will result in a rotational component of velocity given by Vrotr ,rot= , for r~R, (1) Rwhere Vrot is the peak rotational velocity found at thecore radius R. In the potential flow region, vorticity iszero and the rotational component of velocity is givenby VrotR Urot-- , for r>R. (2) rEquations (1) and (2) describe a Rankine combinedvortex circulation. Likewise, within the core region, a field of pure divergence will result in a radial component of velocitygiven by Vrad r Vr,d-- , for r~<R. (3) RIn the potential flow region, divergence is zero and theradial component of velocity is given by VradR Vrad -- for r > R, (4) r where Vrad is the peak radial velocity. Equations (1)-(4) compose the circular model. Ex amples of the circular model are provided in Fig. I for pure rotation (Vrot ~ 25 m s-~, Vrad = 0 m s-l) androtation and divergence (Vrot = 25 m s-l, Vrad = 25m s-~) where R = 2.5 km. Parameter values are selected for illustrative purposes, but are comparable tothe mesocyclone average values reported by Burgessand Lemon (1990), and are typical of a matureOklahoma mesocyclone (Burgess et al. 1982). Forsimplicity, it is assumed herein that the divergence andvorticity fields are identical in size and overlaid inspace. Velocity vectors for each model flow are providedin the top frames of Fig. 1. The large circle identifiesthe boundary between the inner and outer flow regimesand has a radius defined as the mesocyclone core radiusR. The next frames down in this figure depict the Doppler velocity profiles along the dashed and solid lines,AB and CD, respectively, in the Doppler frames immediately below. Doppler velocity Vvov is approximated in the figure as the y component of the twodimensional velocity in Cartesian space. Effects ofbeam filtering and beam angle variation have not beenincluded in this figure. The idealized radar is locatedtoward the bottom of the page. These simple examplesare intended to highlight elementary aspects of the circular model that are observed in single-Doppler data.Doppler velocity contours are evenly distributed between the peaks of Doppler velocity. Flow away fromthe radar is shaded. All similar presentations herein usethe same conventions depicted in this figure. In the circular model, vertical vorticity and horizontal divergence are uniform in the core region (see thelower two frames of Fig. 1 ) and are given by 2Vrot ~= R' (5) 2Vrad D= Rcore region of the Fig. 1 example,Within the= 0.02 s The parameters V~ot and V~a may be assessed fromsingle-Doppler data as (v, - v~) cos(O.) 2 (V, - V,) sin(Or) Vrad = , 2where V, and V, are the inco~ning and outgoing peaksin VDop, and the orientation angle O~ of the velocitypeaks is the orientation angle of the axis between thevelocity peaks relative to the orthogonal to the radarbeam that bisects that axis. Here, Op is defined as 0- forvelocity peaks oriented normal to the bisecting radarbeam, varies from +90- to -90-, and is positive (negative) for counterclockwise (clockwise) rotation ( standard mathematical convention). The orientation angleof the velocity peaks provides an indication of the ratioof divergence to vorticity within the core region: Op= tan-'(~). (6) Another indicator of D/~ is the orientation 4> of thezero Doppler velocity contour (zero isodop). Here, qbis defined as 0- when oriented along the radar beamthat bisects the velocity peaks, and has the same limitsand follows the same rotation convention as 0~. In acircular flow, (3 = 0.. In Fig. 1, 0. and & are 0- (purerotation) and 45- (equal rotation and divergence). One consideration for mesocyclone assessment viasingle-Doppler data is radar resolution relative to mesocyclone size, measured as the ratio of beamwidth tocore radius (BW/CR). Brown and Lemon (1976)2194 JOURNAL OF APPLIED METEOROLOGY VOLUME35found that as the range to a vortex increases, the detected peaks in Doppler velocity are reduced and theapparent vortex size increases, resulting in a significantreduction in the estimated shear across the vortex. A different perspective on the range problem is aspect ratio given by 2 CR sin(/~0)/BW, where/~0 is theangular half-power beamwidth (Wood and Brown1992). As range to a vortex is decreased, the vortexsubtends a larger azimuthal sector of the radar-viewingcircle. At relatively close range, aspect ratios greaterthan 1, the observed Doppler velocity field is decisivelyskewed, with the velocity peaks shifted spatially towardthe radar. As Wood and Brown (1992) explain, thisskewing can result in a smaller vortex appearance, butbecomes significant only when the radar range is of theorder of the core dimension, or less. Resolution effects can be estimated by simulating theradar sampling process with the pulse power densitypattern approximated as a Gaussian distribution in azimuth and elevation and a step function in range. TheWSR-88D radars have a 1- half-power full beamwidthand quantize the velocity data at 1- intervals in azimuthand 0.25-km intervals in range (Crum and Alberty1993). Figure 2 shows the effect of range on the appearance of a circular flow with D/~ of 0 and 1 forBW/CR values of 0.15, 0.5, 1.0, and 1.5. The extentof the core region flow is indicated by the overlaid circle. Intervals .of 1- beamwidth are indicated by theoverlaid straight lines. The central beam is coincidentwith the center of the flow in these examples. The size of the core region is indicated by the distance between the peaks of Doppler velocity. It is alsopossible to infer the two-dimensional shape of the flowfrom the inflections in the contours of the radial velocity field (Fig. 2). The inflections indicate the ring. ofmaximum wind, inside which the winds decrease rapidly toward the flow center, while outside they decreaseat a more gentle rate. At relatively high resolution, BW/CR ~< 0.5, the inflections provide a good indication offlow shape and size. At lower resolution (BW/CR> 0.5), the apparent shape becomes distorted becauseof beam filtering. This effect is depicted by the dashedline for BW/CR = 1.5 and D/~ = 1 (Fig. 2). On average, the accuracy of vortex diameter and shape determination is proportional to the beamwidth. Resolution degradation distorts the shape of the velocity contours within the core region. The velocity contourswithin the core region are straight for relatively highresolutions (BW/CR ~< 0.5), concave for nondivergent flow (D/~ = 0) for BW/CR >~ 1.0, and S-shapedfor divergent and rotational flow (D/~ = 1) for BW/CR = 1.5. The orientations of the velocity peaks (Op) and thezero isodop (40 are also affected by resolution whenD/~ ~= O. Estimates of Op are more sensitive than thoseof 0 to radar resolution. In the D/~ = 1 examples, Opis underestimated by -13- at BW/CR = 1.5. It is interesting that for BW/CR -- 0.5 there is an overestimate of 0, by 13- due mostly to the placement of theradar beam relative to the true peak velocities. By comparison, estimates of ~b are overestimated from 1- to 8-for the same range of BW/CR. In summary, radar resolution impacts the apparentstructure and, therefore, the interpretation of even simple flow structures. These structures are enlarged anddistorted, and shear magnitudes are decreased with reD/~=OD/~=IBW /CR = 0.5 - ~:~ ~:~BW / CR = 1.0 "i BW / CR = 1.5~~!i~i~i~i,lii~FIG. 2. Contours of VDop for the circular flow in Fig. 1 for BW/CR of 0,15, 0.5, 1.0, and 1.5. , WSR-88D characteristics are simulated (refer to text).DECEMBER 1996 DESROCHERS AND HARRIS 2195duced resolution. Mesocyclone intensities are thereforesignificantly underestimated.3. Observations of the Del City mesocyclone from two radars Real mesocyclones are obviously more complexthan the simulated structures of section 2. To illustratethis we have chosen the well-studied Del City mesocyclone of 20 May 1977, for which there are data fromtwo Doppler radars. This storm produced two tornadoes, rated F2 and F3 in intensity. The low-level mesocyclone (1-kin elevation) 14 rain prior to the firsttornado is examined. A detailed description of thestorm structure at this time (1826 CST) as determinedfrom multiple Doppler radar data is provided in Brandes (1984). The observations from the National Severe StormsLaboratory (NSSL) Norman and Cimarron 10-cmDoppler radars are shown in Fig. 3. Note that for consistency with the simulations the radial velocities havebeen adjusted such that the mean of the peak values iszero. The design characteristics of these radars are verysimilar (Ray et al. 1977): both have a 1- half-powerfull beamwidth, for example. The Norman and Cimarton data are quantized in 1- and 0.5- azimuthal increments, respectively, and the datasets were collectedwithin 30 s of each other. The average height of theanalysis domain for each observation is 1 km AGL. The mesocyclone center relative to the Norman and Cimarron radars is 15- and 35-km range, respectively. Of the two observations, it would be expected that a larger apparent shear would result from the Norman data because of the associated finer resolution (Brown and Lemon 1976) and larger aspect ratio (Wood and Brown 1992) as compared to the Cimarron observa tions. However, the shear measured in the Norman ra dar data is 0.0055 s-x, which is smaller than the 0.0065 s-x for Cimarron. The diameter of the meso cyclone core region, the distance between the peaks of Doppler velocity, is approximately 5.0 km for both ob servations. Other inconsistencies become apparent when the in ternal structures of the mesocyclone for the two radars are examined. In each of the Norman and Cimarron observations, a line joining the velocity peaks would be oriented approximately orthogonal to the central beam (0p = -3-), which is indicative of a nondivergent rotational flow (D/~ m 0). As discussed in section 2, in a circular flow, the orientation of the zero isodop is another measure of the ratio of divergence to vorticity within the core. However, for these two radar datasets the orientation of the zero isodop is very different. In the Norman observation 4> ~ 15-, but for the Cimarron observation th ~ -35-. These orientations would sug gest divergence to vorticity ratios of D/~ = 0.25 and D/~ = -0.7, respectively, which conflict with each other and with that from Figure 3 also presents estimates of vertical vorticityand horizontal divergence as determined from a dualDoppler analysis of the Cimarron and Norman data.There are several structural characteristics to the mesocyclone indicated in the vorticity field that are notrepresented by the circular model. As Brandes (1984)points out, the Del City mesocyclone at this time ( 1826CST) was elongated (determined by the shape of theO.Ol-s-x vorticity contour). Also, vorticity is greatesttoward the southern end of the flow and exceeds0.018 s-x (1-km spatial average). The core regioncomprises primarily weak convergence. Collocatedwith the vorticity maximum is a region of enhancedconvergence of -0.007 s-x. West of these maxima,there is a region of moderate anticyclonic vorticity(-0.012 s-~) and divergence (0.008 s-I) that Brandes(1984) associated with downdraft air into the rear ofthe storm. In terms of the single-Doppler flow, thesefeatures lie in what is traditionally referred to as thepotential flow region, where vorticity and divergenceare assumed to be zero. Interpretation of the Del City mesocyclone as a circular flow results in an underestimate of peak vorticityby 30%-40% and an underestimate of peak divergenceby 90%. The errors are noteworthy considering theclose range and high resolution of the data. It is alsoimportant to note that the Del City mesocyclone is notunique; a similar vorticity structure has been assessedfrom multiple-Doppler analysis for other tornadicstorms (e.g., Brandes 1984; Brandes 1993; Johnson etal. 1987) and from numerical models (e.g., Klemp andRotunno 1983; Wicker and Wilhelmson 1993). It isthus logical to develop a model to account for theseobserved variations in vorticity and divergence structure to learn if an improved assessment of single-Doppler data is possible.4. The elliptical model An elliptical model for the flow within a mesocyclone has been developed that permits variability inlengths of axes and vorticity and divergence structures.The effects of each of these variations upon singleDoppler observations will be discussed. This model isa logical extension to an already well-proven circularmodel. The implications of this new model are not onlyin terms of improved interpretation of the overall mesocyclone structure but also in terms of assessments ofthe flow structure within the core region. Variations invorticity structure within mesocyclones could providea large payoff through the improved prediction of severe weather events.a. Nondivergent elliptical fiow The model described here is similar to the circularmodel in that it has two flow regimes that result in aring of maximum winds. In the low-altitude ( 1 km) Del2196 JOURNAL OF APPLIED METEOROLOGY VOLUME35 Cimarron radar-20 DBNorman radarNvelocity vectorsvertical vorticityhorizontal divergence FIG. 3. Dual-Doppler radar data analyses for 20 May 1977 Del City, Oklahoma, mesocyclone at 1826 UTC. Cimarron and Norman Dopplerradar views (second row of frames) at elevation angles of 1.8- and 6.0-, respectively. Contours are at 4 m s-~ intervals. For display purposes,the mean value of the velocity peaks is subtracted and the resultant positive values are shaded. Horizontal velocity vectors, vertical vorticity,and horizontal divergence (bottom frames) are determined from dual-Doppler analysis. Contours of vorticity and divergence are at 0.005 s-~intervals, and posit~ive values are shaded. The overlaid ellipse represents a best-fit estimate of the flow as discussed in section 4c. Profiles ofVDop along lines AB and CD are shown in the top frames.City mesocyclone, the shape of this ring can be described by an ellipse. In cylindrical coordinates, theradius of maximum wind for an ellipse is given by a2b2 ]0.,.7R{~} = b2cos2(-) + a2sin2(~) ,where a and b are the reference major and minor ellipseradii, and e is the ellipse-relative angle that has a valueof 0- along the a axis (see Fig. 4). At e = (0-, 180-),R ----- a. At e = (90-, 270-), R mb. We define thecomponents of the model flow to be oriented paralleland orthogonal to the ring of maximum winds. Flowthat is everywhere parallel (normal) to the ring of maximum winds is nondivergent (nonrotational). Note thatbecause R varies as a function of e, all parameters dependent on R (e.g., Vrot and Vrot) will also be expressedas functions of e. In a rotational nondivergent flow, the mass flux between 0 and R { ~ } is given by phR { e } ~rot { ~ }, where pis air density, h is vertical thickness, and ~ot { e } is theaverage rotational velocity between 0 and R { ~}. If pand h are constant, thenR{e}~rot{~} = a~rot~ = b~rotb,where V~o~ and V-~o~o are the average rotational velocitiesacross the a and b axes of the ellipse.DECEMBER 1996 DESROCHERS AND HARRIS 2197VrotF~G. 4. Schematic of geometry used for the elliptical model. Refer to section 4a. At e = 0-, 90-, 180-, and 270-, the tangent to theellipse is normal to the radius and ~rot { e } is equal to~r { e}, the average tangential velocity between 0 andR { e }. In general, Trot{e} ~--- ~T{e} sin(a),where a is the angle between the radial at angle e andits associated tangent to the ellipse. The angle a is afunction of e and the elongation factor (a/b): a=tan-I ~:)tan(e)] +6,as presented in Fig. 4. In the circular Rankine combined vortex model, velocity varies linearly with radius within the core andinversely to radius outside the core. As a starting point,this relationship is maintained for elliptical flows. Thetangential velocity Vr within the core region (0 ~< r<~ R { e } ) varies according to VT{e}r Vrot { e} r vT- R{e~ - R{e} sin(a) ' (7)and, in the outer flow region (r > R { e } ), by u~ - - , (8) r r sin(a)where Vr { e } is the peak tangential velocity at the coreradius (r = R{e}). The parameters V~ot{e} andVr { e } are defined by V~o, { e } = 2V~o~ { e } and Vr { e }= 2~r{ e}, where a linear velocity variation across thecore region is assumed. The notation Vr~ ~ Vr { e = 0-,180- } and Vr, ~ Vr{ e = 90-, 270- } is followed herein.Within the core region where the velocity distributionis defined by Eq. (7), the vo~icity is unifo~. Figure 5 presents example elliptical flows at variousorientations to the radar with the flow characteristicsselected for illustration. The elongation factor is a/b= 2.0, which co=esponds to an eccentricity of 0.866.The minor axis radius and peak rotational velocity ~eb = 2.0 ~ and Vr~ = 32 m s-~, respectively, so thatvertical vorticity in the core region is the same as inthe circular model example (Fig. 1 ), 0.02 s-~. The ellipse orientation angle 0E is defined as 0- when the minor axis of the flow is oriented in the same direction asthe radar beam that bisects the flow. Ellipse orientation can significantly impact the appearance of a flow. Consider, for example, the variationin orientation of the zero isodop (~ with ellipse orientation. In a circular flow, ~b is always normal to thechord connecting the velocity peaks. In this elliptical,nondivergent flow, ~b is normal to the chord connectingthe velocity peaks only when the radar beam is alignedwith a principal axis. Here, ~ varies between +31- and-3l- as 0E varies from 0- to 135-. In contrast, 0p isinvariant with 0~, being 0- for all ellipse orientations.In Fig. 5, the fields for 0~ = 90- approximate the Norman radar observations, while those for 0~ = 45- approximate the Cimarron observations (Fig. 3). Brown and Lemon (1976) discuss how degradingresolution reduces the magnitude of shear measured between the velocity peaks. The measurement of shear inan elongated flow can be comparably impacted by theorientation of the flow relative to the radar. In an elongated flow, the shear between the peaks of Dopplervelocity varies with ellipse orientation, which will influence the assessment of intensity. In the examples inFig. 5, azimuthal shear between the velocity peaks varies by a factor of 4, from 0.004 s-~ for 0~ = 0- to0.016 s -~ at 0E = 90-. Figure 6 depicts the variation inazimuthal shear for nondivergent flows with ellipse orientation (0E) for a range of elongation factors (a/b).Radar resolution effects are not considered. For eachflow defined by (a/b), shear is normalized to the peakvalue at 0~ = 90-. Shear can be underestimated by 75%for an elliptical flow with an elongation factor of a/b= 2.0. Even a moderate flow elongation can have aconsiderable impact on the detected shear; shear maybe underestimated by 36% when the elongation factoris only a/b = 1.25. Azimuthal shear is one of the primary quantities usedto identify a mesocyclone and estimate vorticity fromsingle-Doppler radar (Donaldson 1970; JDOP Staff1979). Therefore, the dependence of estimated shearon ellipse orientation can impact the detection of mesocyclones. The formulation used in these studies isthat vorticity equals twice the azimuthal shear measured between'the velocity peaks [ Eq. (5)], which assumes a circularly symmetric flow and uniform vorticity within the core region. However, for noncircularflow, vorticity determination requires a measure ofshear in two directions. Figure 7 shows ratios of apparent vorticity when shape is not considered to truevorticity. The magnitude of the estimation errors depends on the ellipse elongation factor and orientationto the radar. For example, azimuthal shear for a flowwith a/b = 2.0 implies a vorticity magnitude that is40% of the actual at 0~ = 00, but 160% of the actual at0E = +90-. There is a unique relationship at 0E = +--45-,2198 JOURNAL OF APPLIED METEOROLOGY VOLUME35VDop(m/s)VDop0E = 0- 0E = 45- 0E = 90- 0E = -45-40~/ ~~~B],-'~ B /^'~-40 FIG. 5. Elliptical flow field characteristics at various orientations where b = 2.0 kin, alb = 2.0, and Vr~ = 32 m s-~, The flowis nondivergent. Top frames show horizontal velocity vectors for pure rotation. The large ellipse indicates the boundary betweeninner and outer flow regimes. Simulated Doppler velocity fields VDop (middle frames) are for a radar located at bottom of figure.Doppler velocity contours are every 5 m s-~ with positive values shaded. Profiles of VDop along ~-~ and ~-~ are shown in framesabove. Vertical vorticity (bottom series of frames) is contoured at 0.005 s-~ intervals with positive values shaded. Radar resolutionis not considered.where vorticity is equal to twice the azimuthal shear,regardless of elongation factor. Radial shear within the core region of a circular flowindicates the presence of divergence. However, in anelongated nondivergent flow the radial shear may benonzero within the core region. For example, in thenondivergent elongated flows depicted in Fig. 5, themagnitude of radial shear is 0.006 s -~ for 0E = 45-.b. Vorticity in the outer fiow region One characteristic of the circular model is that vorticity is zero in the potential flow region (Fig. 1 ). However, in an elliptical flow there is structure to the vorticity field outside the core (Fig. 5). For an elongatedcyclonic flow, positive vorticity extends into the outerflow region along the major axis of the flow, and acouplet of anticyclonic vorticity is aligned with the minor axis. This vorticity structure results from the rotational velocity outside the core being constrained tovary as 1/r [see Eq. (8)]. Although the flow of thecore region is elliptical, flow outside the core becomesincreasingly circular with distance from the vortex center. A similar anticyclonic vorticity couplet and elongated positive vorticity structure is seen in the Del Cityanalysis (Fig. 3) where there is a region of anticyclonicvorticity in the same general area indicated by the elliptical model. The relative magnitude of the anticyclonic vorticityin the potential flow region compared to the cyclonicvorticity in the core increases with elongation factor(not shown). For a given elongation factor, the magnitude of vorticity would be diminished (increased) ifvelocity tapered off at a rate less (greater) than 1/r inDECEMBER 1996 DESROCHERS AND HARRIS 21991.000.750.500.251.5 2.00o -+10o :t:20- -I-30- :t:40- :t:50o _+60- _+70- +80- _+90- ellipse orientation angle (0E) FIG. 6. Ratio of detected shear to peak shear versus ellipse orientation angle 0e for several elongation factors a/b for a pure rotationalflow. Radar resolution is not considered.Eq. (8). In all the discussions herein a rate of 1/r isused.c. Uniform vorticity and divergence The relationships that govern divergent flow areanalogous to those of rotational flow. Equations (7)and (8) can be rewritten to represent divergent flow.Inside the core region (0 ~< r ~< R{ e}) the normalcomponent of velocity to the ellipse is represented by VN{e}r Vrad { ~} r vN- R{e~ - sin(a)R{e} ' (9)where V~v { e } and Vr~d{ e } ~e, respectively, the peaknodal and peak radial velocity at r = R { e}. The notation V~ ~ V~ { e = 0-, 180- } and V~ ~ V~ { e = 90-,270-} is used. In the outer flow region (r > R { e } ), r r sin(a) Figure 8 presents an elliptical flow at various ohentations to the radar where Vs~ = Vr~ and the fields ofvo~icity and divergence ~e equal (D/( = 1 ). Of issueis how the parmeters 0, and & relate to the shape ofthe flow and the magnitudes of vorticity and divergence. In elliptical and circul~ flows alike, 0e is an indicatorof the relative magnitude of divergence to vo~icitywithin the core region as described in Eq. (6). Thisrelationship is supposed by Figs. 5 and 8 where D/~= 0 and D/~ = 1, respectively, and 0, = 0- and 0,= 45-, respectively, regardless of the ohentation of theflow to the radar. In general, as 0, increases, divergenceincreases relative to vorticity. The orientation of the zero isodop varies greatly with0E: ~b = 76-, 45-, 14-, and 45- for 0E = 0-, 45-, 90-, and-45-, respectively, for the Fig. 8 examples. In general,~b is given by [ a2 tan( Op - OE) ] q~ = tan-~ b2 +where ~b is constrained between +90- and -90-. For agiven elongation factor and vorticity to divergence ratio, the ellipse orientation for the greatest departure angle is given by OE(~)max)~Op--+ tan-~(~) -Figure 9 shows qb - Op as a function of 0E - 0p for arange of elongation factors a/b. The parameters of ~b and 0~ are readily estimatedfrom single Doppler data. From these two parametersthe range of possible combinations of ellipse orientation 0E and elongation factor a/b can be determinedfrom Fig. 9. However, from this approach, these parameters cannot be determined uniquely. Fortunately,independent estimates of mesocyclone orientation andelongation factors may be obtained by fitting an ellipseto the Doppler velocity field. As discussed in section2, it is sometimes possible to infer the outline of themesocyclone core region from the inflections in thecontours of the Doppler velocity field. The use of Fig.2.0;>O'Ec ~ 0.5elongation factor (a/b) 0o -+10o _+20- _+30- _+40- _+50- -+60- _+70- _+80- _+90- ellipse orientation angle (0~) FIG. 7. Ratio of apparent vorticity to actual vorticity versus ellipseorientation angle 0e for several elongation factors a/b for a pure rotational flow. Radar resolution is not considered.2200 JOURNAL OF ,APPLIED METEOROLOGY VOLUME35VDop(m/s)~/DopD0E = 0- 0E ---- 45- 0e = 90- 0e = -450-40Fro. 8. Same as Fig. 5 except that DI~ = 1. Bottom series of frames shows horizontal divergence contoured at 0.005-s-~ intervals with positive values shaded.9 would serve to validate estimates derived from thecurve fitting technique. The Cimarron and Norman data provide a test bedfor the above approach. An ellipse of a/b := 2.7 and0e = 52- provides a reasonable visual fit to theCimarron data. From Fig. 9, these parameters and themeasured 0, of -'3- yield a (4) - 0,) of -29-, whichis within 3- of that actually measured. The visual fitto the Norman data yields a/b = 2.8 and 0e = -76-.With the same approach as above, this analysis alsoyields a difference in 4) - Op within 3- of thatactually measured. When 0e from the two visual fits areadjusted for radar viewing angle differences, the estimates for mesocyclone orientation are within 8- of eachother. The estimates for a/b are within 4%. Since this mesocyclone was viewed from two radars,there is an alternative approach to determine a'/b and0E, where the measured parameters of 4) and 0, fromeach radar are combined. It is also necessary to knowthe azimuth from each radar to the center of the meDECEMBER 1996 DESROCHERS AND HARRIS 220160-50-40-30-20-,~ I0- ~ 0-~ _10o! -20- -300 -40~ -50* -60- _90- -70- -50- -30- _I0o 10o 30- 50- 70- 90- (0~- 0p) FIG. 9. Departure of the zero isodop 4~ from velocity peaks orientation 0p versus departure of ellipse orientation angle 0~ from 0,, forseveral elongation factors (a/b).socyclone. The difference between these azimuths(AOE) is the viewing angle difference, which may berepresented aswhere 0ec,m and 0ENos. are, respectively, the ellipse orientation angles relative to the Cimarron and Norman radars. From the estimations of 4~ and 0, from the data foreach radar, it is possible to determine a range of valuesfor a/b and 0e - 0p for each radar. For example, a (~b-- 0p)Nor ~ 18- for the Norman radar yields ranges of1.3 < a/b < 3.0 and -70- < 0E - Op < -3.0-. Similarranges can be found for the Cimarron radar data. If thetwo radars are observing the same feature, then the a/bmust be the same for both observation sets. Also, theviewing angle difference A0e together with the individual 0, values constrain a/b, Oec~m, and 0~Nor to uniquevalues. Therefore, with the estimates of tb, OF, and= 120; Fig. 9 yields an ellipse elongation factor of a/b= 2.7 and orientation angles of 0Ecim : 48- and=--72-. These values compare very favorably withthose determined from each of the datasets as cited inthe previous paragraph. An ellipse with these characteristics is overlaid onto the Del City mesocyclone data inFig. 3 and appears to provide a good fit to the observations. It is noteworthy that the fitted ellipse closelymatches the outline of the 0.005-s-2 vorticity contour. From the estimates of a/b, 0e, and shear betweenthe velocity peaks, an estimate of the vorticity withinthe core region can be determined from Fig. 7. As discussed in section 3, shear between velocity peaks isestimated at 0.0065 s-~ and 0.0055 s-~ for the Cimarron and Norman observations, respectively. For a circular flow these shears translate into vorticity magnitudes of 0.013 s-t and 0.011 s-~. From Fig. 7 for theCimarron observation, a/b = 2.7 and 0eC,m = 48- correspond to an underestimate of vorticity by a factor of1.1. Likewise for the Norman observation, where OL-~o~= -72-, a circular assumption results in an underestimate of vorticity by a factor of 1.6. Therefore, whenthese corrections are applied, the estimates of vorticityfrom the Cimarron and Norman observations are, respectively, 0.014 s-~ and 0.018 s-~. The correctedNorman estimate is identical to the dual-Doppler estimated peak value, while a modest improvement isnoted for the estimated value for the Cimarron data.However, these estimates assume uniform vorticitywithin the core region, which is not the case in the DelCity mesocyclone (Fig. 3). The effects of nonlinearitywill be addressed in section 4e. From this analysis it is demonstrated that mesocyclone flow shape is reasonably approximated by an ellipse and that a reasonable assessment of the mesocyclone flow structure can be determined from singleDoppler data.d. Effects of radar resolution Radar resolution impacts the estimation of vortexsize (Brown and Lemon 1976) and peak velocity magnitudes. Examples of how the simple elliptical flow depicted in Figs. 5 and 8 would appear at various resolutions (BW/CR) and orientations are shown in Figs.10 and 11, respectively. Of concern is the impact ofresolution on the determination of flow shape and themeasurement of tb and 0,. Characteristics of the WSR88D radar, namely beamwidth and sampling density,are simulated in these examples. The model flow iscentered on the radar beam that bisects the flow (centralradar beam). An ellipse representing the true core region shape is overlaid for reference. Core radius for anelliptical flow is defined as one-half the length of thechord across the ring of maximum winds, where thechord passes through the flow center at an angle orthogonal to the central radar beam. At BW/CR ~< 0.5, which corresponds to rangeswithin 115 km, 72 km, and 57 km at 0~ = 0-, 45-, and90-, respectively, for both figures, contour inflectionsprovide a reasonable outline of the core region. Atgreater values of BW/CR, the inflections trace a shapethat is more circular than actual, especially at 0E = 45-and 90-. Also, the velocity contours are no longerstraight lines within the core region. This is especiallyapparent at BW/CR = 1.5 and 0s = 45-. Of additional concern is the variation in the orientation of the velocity peaks with radar resolution. Thereis no divergence in Fig. 10, and 0p should be 0- according to Eq. (6), but it varies from -5- at BW/CR= 1.0 to +10- at BW/CR = 1.5 for 0e = 45-. In Fig.11, where the magnitudes of divergence and vorticityare equal within the core, Op should ideally be 45-, but2202 JOURNAL OF 'APPLIED METEO,ROLOGY VOLUME350E = 90~0E =45-0E = 0-BW / CR = 0.15 :~13 BW/CR=0.5'}}i~iiiBW / CR = 1.0BW / CR = 1.5 ,:.!Fro. 10. The same velocity field in Fig. 5 sampled with BW/CR = 0.15, 0.5, 1.0, and 1.5 for O~ = 0-, 45-, and 90-.The thin straight lines indicate the bounds of the 1- radar beam. Contour intervals are every 5 m s-~.at BW/CR = 1.5 it varies considerably from 130- at 0~= 90- to 85- at 0~ = 0-. The measurement of ~ does not appear to be as sensitive to radar resolution as O,. In the Fig. 10 examples,qb is within 1- of its theoretical value for each orientation for the various BW/CR ratios. There is more variation seen in ~ when there is a component of divergence to the flow. In Fig. 11, there is a 15- variation in~b across the range of BW/CR ratios for 0~ = 90-, a20- variation for 0~ = 45-, but only a 5- variation for0~ = 0-. It is ironic that the measurement of 0p should be impacted more than 4> by ellipse orientation when radarresolution is considered, since it is independent of ellipse orientation when radar resolution is not considered. It appears that resolution effects restrict the assessment of the size, shape, and orientation of mesocyclone-sized flows to 100 km or less. Beyond thisrange, uncertainties in the determination of'4) and 0,make the determination of flow shape and orientationto the radar quite difficult.e. Nonuniform vorticity Earlier discussions deal with the interpretation offlow shape in single-Doppler data when vorticity anddivergence are uniform within the inner flow (mesocyclone core) region. However, the vorticity and divergence fields within the Del City mesocyclone areclearly not uniform as indicated from the dual-Doppleranalysis in Fig. 3. When both vorticity and divergenceare uniform the azimuthal and radial profiles of Doppler velocity within the core region are linear, and theisodops within the core region are straight lines (seeFigs. 1, 5, and 8). In the Del City mesocyclone theisodops were S-shaped and the Doppler velocity profilebetween the velocity peaks was approximately sinusoidal (Fig. 3 ). The effects of these types of variationsacross the core are now examined. This section addresses variations in the tangential velocity. The tangential velocity for 0 ~< r <~ R { e } is assumedto vary as a single harmonic according toVr- R{e~ + krVr{~} sin , (10)where kr is a weighting factor. The first term in Eq.(10) comes from Eq. (7) and results in uniform vorticity within the core region. The second term in Eq.(10) adds a sinusoidal variation to the tangential velocity. Vorticity is nonuniform within the core regionDECEMBER 19960E =90-0E =45-0E = 0-DESROCHERS AND HARRISBW / CR = 0.5BW / CR = 1.0 2203BW / CR - 1.5for kr q= 0. For kr > 0, Ovr/Or is greatest at the flowcenter and is at a minimum at r = R { e }. For kr < 0,~vr/Or is greatest at r = R { e} and at a mini~num at theflow center. Figure 12 provides examples of the flow for kr = 0,0.2, and -0.2, respectively. There is no divergence inthis example. Results are shown for 0~ = 45- to compare with the Cimarron radar view of the Del City mesocyclone. For kr = 0.2 (middle column), vorticity isorganized in a bull's-eye and the isodops for the Doppler velocity field are slightly concave-shaped. For kr= -0.2, vorticity is organized in a ring and the isodopsare convex-shaped. Of these examples, that for kr = 0.2most closely resembles the vorticity structure, isodopshapes, and Doppler velocity profile for the Del Citymesocyclone (Fig. 3). In Fig. 12 the velocity peaks and the shear measuredbetween them does not change when the velocity profile in the core region is altered from linear to sinusoidal. However, for kr = 0.2 and kr = -0.2 the maximumvorticity values are, respectively, 60% and 50% largerthan that for the linear velocity variation (kr = 0). Thisfurther indicates that shear measured between the velocity peaks may be an inaccurate indicator of mesocyclone circulation intensity. Note that for this model,the areal average of vorticity within the core region isthe same for the linear and both of the sinusoidal variations, namely 0.02 s-~. The shear between the velocity peaks is proportional to the average vorticity withinthe mesocyclone core region, but not necessarily thepeak value of vorticity. Assessment of the peak vorticity requires a description of the flow structure withinthe core region.f. Nonuniform vorticity and divergence Isodop structure interpretation is further complicatedwhen fields of vorticity and divergence are both nonzero and not constant across the core region. This is amore physically reasonable situation than the one described in the previous section. A general descriptionof the normal velocity to the ellipse for 0 ~< r ~< R { e }is given by V,v - g{e~ + k~vV, v{e} s~n , (11)where k~v is the divergent velocity perturbation weighting factor. The first term of Eq. (11 ) comes from Eq.(9) and results in uniform divergence within the coreregion. The second term produces the sinusoidal variation. Figure 13 provides examples for 0p = 45- and arange of kr and k~ combinations.2204 JOURNAL OF APPLIED METEOROLOGY VOLUME35kT =0. kT =0.2 kT =-0.2Voo~ o ............................. -"-' FiG. 12. The VDop profiles, Voop contoured plots, and vo~icity fields for line~ (kr = 0) andnonline~ (kr = 0.2, -0.2) velocity vmations within ~e core region where a/b = 2.0, b = 2.0kin, and Vr~ = 32 m s-~. The flow is nondivergent. Here, 0~ = 45- to approximate the view fromthe Cima~on Doppler radar. Contour levels are as in Fig. 5. Internal variations in either one or both fields produce a nonlinearity in the isodops across the core region. In the previous section it was shown that convexor concave isodops result when there is a vmfation inone field while the other is zero (Fig. 12). A similarisodop structure will be produced when both fields arenonzero and have internal variations that are in phase.Two examples of in-phase variations are shown in Fig.13 for (kr, kt~) = (0.2, 0.2) and (-0.2, -0.2). Whenthe fields of vorticity and divergence are out of phase,the VDop profiles become sinusoidal in the core region.In the examples for (kr, kN) = (-0.2, 0.2) and (0.2,-0.2), the VDop profiles and the core region isodopsbecome S-shaped or backward S-shaped. The shape ofthe isodops in all cases is related to the variation of D/~ across the mesocyclone core. An S-shaped isodopstructure is interpreted as divergence increasing relativeto vorticity toward the mesocyclone center. The backward S-shaped isodops indicate that divergence is decreasing relative to vorticity toward the center of theflow. Linear, convex, or concave isodops indicate thatD/~ is invariant across the mesocyclone core. A nonlinear zero isodop also indicates that Op maynot be representative of the peak values of divergenceand vorticity. An orientation angle of the Del City mesocyclone velocity peaks of 0, = -3-, measured fromFig. 3, would, from a traditional assessment, suggestthat divergence within the core is approximately equalto zero. However, the dual-Doppler analysis in Fig. 3shows that the peak divergence is about 40% of thepeak value of vorticity. This discrepancy is probablydue to considerable fluctuations in the divergence fieldon a scale smaller than the mesocyclone core, as seenin Fig. 3. The general Del City mesocyclone divergencestructure can be approximated with the second term ofEq. ( 11 ), the perturbational variation of VN. Figure 14provides examples for the perturbational variation of v'Nwhere k~ = 0.2. The magnitude of divergence increasestoward the center of the flow, but the areal average ofdivergence is zero within the core region as indicatedby Op = 0-. In each of the examples of Fig. 14, the isodops inthe core region are S-shaped. Note, for the first column(kr = 0), vorticity is uniform in the core region. Theshape of the isodops indicates how the fields of vorticity and divergence change relative to each other, butdoes not indicate the structure of an individual field. Ineach case, the S-shape indicates that D/~ increases toward the center of the modeled flow. Similarly, decreasing D/~ toward the center yields backward Sshaped isodops (not shown). In all preceding examples, the azimuthal profile ofDoppler velocity through the core region is linear andthe vorticity field is uniform (e.g., Figs. 5 and 8). However, in the first column (kr = 0) of Fig. 14 the profileacross AB is a sinusoid even though the vorticity isDECEMBER 1996 DESROCHERS AND HARRIS 2205 kT =0 kN=00(m/s) -40VDopDkT =0.2 kT =-0.2 kT =-0.2 kT =0.2ks =0.2 kN =-0.2 kN =0.2 kN =-0.2 /\ / \ xx.............. / xFro. 13. As in Fig. 12 except that divergence also varies from linear (k~v = 0) to nonlinear (kN = 0.2, --0.2). The top frames depict D/~ through the core region along the line AB for each case.uniform. Therefore, it would appear that azimuthal andradial Doppler velocity profiles through the mesocyclone are not entirely reliable indicators of the vorticityor divergence structure. The second column of Fig. 14, where (kr, kN) = (0.2,0.2), provides the closest match to the vorticity anddivergence fields of the Del City mesocyclone, and thesingle-Doppler appearance is similar to the Cimarronradar view. However, there are only subtle differencesin the shapes of the isodops of the (kr, kN) = (0, 0.2)and (0.2, 0.2) examples, even though they are associated with very different vorticity structures. Vorticityin the core is uniform in the (kr = 0) example, but isboth bull's-eye shaped and 60% greater in magnitudefor (kr = 0.2). It is apparent that the shape of the isodops is insensitive to the internal vorticity structure.In fact, the shape is also insensitive to the internal divergence structure. However, it is extremely sensitiveto the relative variations of divergence and vorticity(D/l). As noted before, when divergence increasesrelative to vorticity toward the core center, the isodopsare S-shaped, and when divergence decreases relativeto vorticity, the isodops are a backward S-shaped. Foroperational radars like the WSR-88D, the general Sshape of the isodops in Fig. 14 can be detected to arange of approximately 100 kin.g. Asymmetric fiows In the preceding discussions the center of the flowcorresponds to the center of geometry. This section addresses the interpretation of asymmetric flows wherethe flow center within the core region is displaced fromthe geometric center. Such a scenario is observed in theDel City mesocyclone, where vorticity is concentratedtoward the southern end of the flow (Fig. 3).2206 JOURNAL OF APPLIED METEOROLOGY VOLUME35 kT=0 ktq =0.2-2kT =0.2 kT =-0.2kN =0.2 kN =0.2Voov o -40// \\DFIG. 14. Similar to Fig. 13 except that divergence is computed by the second part of Eq. (l 1). Conservation of angular momentum reqmres achange in core radius to be offset by an inverse changein rotational velocity. In an asymmetric flow, rotationalvelocity and vorticity will be enhanced where the flowis narrowed. This relationship applies to elliptical andcircular flows alike. Figure 15 provides examples of asymmetric flowwhere the vortex center (C) is displaced in either themajor or minor radii directions to (0.4a, 0b), (-0.4a,0b), (0a, 0.2b), and (0a, -0.2b). The flow orientationis intended to approximate the perspective from the Cimarron radar. There is no divergence in these examples. An asymmetric flow is identified by an asymmetryin the isodop pattern. While asymmetric flows can bereadily identified, the accurate placement of their centers in real data is very difficult. In the model'examplesin Fig. 15, lines AB and CD intersect at the flow center.This flow center lies along a chord connecting the velocity peaks. However, placement of the center alongthis chord is very difficult to gauge because there is noidentifiable signature associated with the flow center.It is possible to get an estimate of the flow center fromthe intersection of the chord connecting the velocitypeaks and ihe isodop corresponding to the mean of thevelocity peaks. The simulation for the vortex center at(0.4a, 0b) in Fig. 15 most closely approximates theCimarron observation of the Del City mesocyclone(Fig. 3). Flow asymmetry further complicates the estimationof vorticity. The model flow in Fig. 15 has the samedimensions and characteristic velocity (Vrb = 32DECEMBER 1996 DESROCHERS AND HARRIS 2207C = (0.4a,0b)C = (-0.4a,0b)C = (0a,0.2b)C = (0a,-0.2b)VDop(m/s)VDop//////-~ ~ l F~G. 15. Flow field characteristics of asymmetric nondivergent flow with the same general characteristics as in Fig. 5. The flowcenter C is shifted along the major or minor axis direction and is represented as a fraction of a and b. Here, 0E = 45- to approximatethe view from the Cimarron Doppler radar.m s -~) as the Fig. 5 examples. However, values of peakvorticity are 0.035 s--~ for the leftmost frames [C= (_0.4a, 0b)] and 0.03 s-~ for the rightmost frames[C = (0a, +0.2b)] in Fig. 15, while vorticity is0.02 s -* in the Fig. 5 examples. This difference in vorticity magnitude is significant given that the measuredshear between the velocity peaks is, respectively, only20% and 10% greater than the con'esponding values inFig. 5. Asymmetries can produce regions of locally enhanced vorticity that are not well represented by themagnitude of shear between the velocity peaks.5. Simulation of the Del City mesocyclone From a manual inspection of the single-Dopplerdata, the Del City mesocyclone is found to have thefollowing characteristics as interpreted via the ellipticalmodel: a/b = 2.7, 0, = -3-, 0 = -35-, and 0~ = 48-for the Cimarron observation, and tb = 15- and 0E= -72- for the Norman observation (section 4c). Thevortex/difluence center is estimated at (0.4a, -0.2b),and the rotational velocity is Vro = 15 m s-~. TheDoppler velocity was shown to vary as a sinusoidacross the mesocyclone core region. With amplitudesof kr = 0.2 in Eq. (10) for vorticity and kN = 3.0 inEq. ( 11 ) for divergence and the other parameters listedabove, a simulation of the Del City mesocyclone hasbeen generated (Fig. 16). WSR-88D radar samplingcharacteristics are used in the simulations. Fields ofvorticity and divergence are computed on scales comparable to the dual-Doppler derived fields (Fig. 3),namely 1 km. The simulation closely approximates the singleDoppler mesocyclone appearance from the vantagepoints of the Cimarron and Norman radars (Fig. 3).There is general agreement in the structure of the simulated vorticity and divergence fields with the fieldsdetermined from the dual-Doppler analysis. Vorticity2208 JOURNAL OF APPLIED METEOROLOGY VOLUME35 Cimarron radar simulation Norman radar simulationVDop / / \ .~B -20 D F~G. 16. As in Fig. 12 but for simulations of the Del City mesocyclone where a/b = 2.7, Op = -3-, Vr~ = 15 m s ~, kr = 0.2, k~= 3.0, and ~e cenwr of the model flow is C = (0.4a, -0.2b). Thesame flow field is sampled for 0~ = 48- and -72- to approximate therespective Cim~on and Norman views, respectively. Range effectsof beam filmfing and angul~ v~iation are inco~orated into the simulations. The simulation for the Cima~on radar represents a range of35 km, while the range for the No~ rad~ simulation is 15 km.is enhanced toward one end of the core region and displays a bull's-eye structure. Divergence increases relative to vorticity toward the center of the flow, whichresults in the characteristic S-shape of the zero isodop. In the outer flow region, there is a couple, t of anticyclonic vorticity that is enhanced to one side in a similar manner as observed in the dual-Doppler analysis(Fig. 3). The dual-Doppler analysis shows a more significant vorticity feature that extends south, then west,from the core, which Brandes (1984) associated withthe gust front. The intent of this study is to focus onthe structure within the core region of the mes, ocyclone.Therefore, gust fronts and other complexities in theouter flow region are ignored in the simulations. Therefore, with parameters measured from the single-Doppler data and a relatively simple kinematicmodel, it is possible to generate a reasonable simulationof a mesocyclone flow field.6. Discussion and conclusions Historically, single-Doppler mesocyclone observations have been interpreted as a circular flow with aRankine combined vortex velocity distribution. Interpretation of the 20 May 1977, Del City, Oklahoma,mesocyclone 'observations from the Norman and Cimarron radars as a circular flow results in an underestimate of peak vorticity by 40% and 30%, respectively,and an underestimate of peak divergence by over 90%.It is shown that these underestimates are due primarilyto the inadequacies of the circular model in representing the mesocyclone structure. In the circular modelmesocyclone assessment is based entirely upon theshear measured between the velocity peaks. It is assumed that vorticity is uniformly distributed in the mesocyclone core region and is zero elsewhere. This isclearly not the case at low altitude for the Del Citystorm (Fig. 3). A new mesocyclone model has been developedwhere mesocyclone flows are approximated as ellipsesof specified elongation factors. Within the mesocyclonecore, a wide range of vorticity and divergence struc-'tures have been specified through simple functions.Nonrotational and nondivergent components of theflow field are generated separately to assess the contribution of each to the simulated single-Doppler appearance. The flow field has been placed at a variety oforientations and ranges relative to the radar to simulatethe single-Doppler appearance. Flow elongation factor and orientation to the radargreatly impact single-Doppler mesocyclone appearance. For the Del City mesocyclone, many of the Doppler velocity structure differences for the Norman andCimarron observations are attributable to the elongatednature of the flow and the different radar viewing angles. In the Del City mesocyclone observations, it hasbeen possible to visually gauge the parameters of flowelongation factor and orientation to the radar from theridges of local maxima in the Doppler velocity field.These parameters govern the orientation of the zeroisodop relative to the velocity peaks. The orientationsof the zero isodop and of the velocity peaks indicate atheoretical range of possible flow elongation factorsand orientations to the radar. These simple flow characteristics should be determinable to ranges beyond100 km for the WSR-88D radar if the minor radius ison the order of 2.5 km or'greater. The assessment of mesocyclone intensity throughthe use of peak-to-peak shear is significantly impactedby flow ghape, orientation to the radar, asymmetry, andinternal velocity structure. For a simple elongated flowwhere vorticity is uniform within the core region, thedetected shear can vary by a factor of 4 with ellipseorientation when the major axis is twice the minor axisDECEMBER 1996 DESROCHERS AND HARRIS 2209length. Asymmetries and internal velocity structuresfurther widen the gulf between the estimated and truevorticities. The most visible effects of asymmetries and internalvelocity structures are seen in the shape of the isodops.This is especially true for the zero isodop, which isindicative of the relative variation of divergence andvorticity across the core region. It is not possible toaccess the absolute structure and magnitude of the individual divergence and vorticity fields. However, it ispossible to retrieve some information on their relativevariability. For example, an S-shaped zero isodop indicates that divergence increases relative to vorticitytoward the center of the flow. The opposite trend isindicated by a backward S-shaped zero isodop. The DelCity observations have an S-shaped zero isodop that isin agreement with the dual-Doppler assessments of vorticity and divergence (Fig. 3). In general, it appearsthat the shape of the zero isodop can be assessed toranges of 100 km for most mesocyclones. It has been shown from multiple-Doppler analysis(e.g., Brandes 1978, 1984; Ray et al. 1981) that mesocyclone vorticity structure undergoes significant evolutionary changes leading to tornado formation. Itwould therefore appear reasonable to monitor the zeroisodop because changes in its shape may signifychanges in the internal mesocyclone structure. Thiswork will be addressed in the future once the automation of the model is completed. Automation will permitthe processing of large quantities of data in order tobetter relate mesocyclone structure as derived from single-Doppler radar to tornadic development. Acknowledgments. 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