VOL. 43, NO. 11 JOURNAL OF THE ATMOSPHERIC SCIENCES 1 JUNE 1986Precipitation Growth Trajectories in a CCOPE StormCHARLES A. KNIGHTNational Center for Atmospheric Research,* Boulder, CO 80307 KEVIN R. KNUPP** Colorado State University, Fort Collins, CO 80523(Manuscript received 12 April 1985, in final form 26 August 1985)ABSTRACTThe growth trajectories of precipitation particles that attain diameters from 0.5 to 2.0 cm are modeled withinthe wind field of a small, relatively steady-state, southeastern Montana thunderstorm. The trajectories are calculatedbackwards, from systematic amys of particles of specified sizes at a level near cloud base. Using a simple Setof criteria for rejecting the obviously impossible trajectories, the patterns of accepted trajectory end-points arecompared with the radar echo patterns. Good agreement lends credence to the qualitative aspects of the trajectories.For a given size of precipitation particle, the method helps one to assign different trajectory types to specificregions within the horizontal plane on which the calculations were started. The relative importance of thedifferent types of trajectories can thus be estimated. Particle origin mechanisms are discussed in terms of theregions in which the trajectories are found to start. The variety of successful trajectories leading to 1 cm andlarger hail in a storm of considerable structural simplicity is noteworthy.Sensitivity tests indicate that the liquid water content is by far the most important specification in thisframework. Ongoing work is directed toward improving this specification and deriving estimates of particleconcentration.1. IntroductionThis is a study of precipitation growth trajectories.Given certain wind and microphysical data on oneparticular storm, what types of trajectories were followed by the particles that reach precipitation size?While this is a reasonable goal for its own sake, sinceprecipitation will not be understood without the particle growth trajectories being determined, there areseveral, more specific purposes of this study. In particular, the methodology of trajectory determination andthe sensitivity to several factors are examined, and theresults are used to deduce something about particleorigins in, the storm and to interpret the shape of thestorm's radar echo.Historically, most growth trajectory studies havebeen concerned with hail. Early studies (e.g., Browning,1963; English, 1973) used highly idealized wind fields.With the advent of multiple-Doppler radar techniquesfor estimating three-dimensional wind fields in somedetail, more complicated and realistic trajectories havebeen described. Paluch (1978) was one of the first touse Doppler wind fields for this purpose, examining* The National Center for Atmospheric Research is funded by the** Present affiliation: Kenneth E. Johnson, Environmental andNational Science Foundation.Energy Center, University of Alabama in Huntsville, AL 35899.the growth of small hail in a wind field obtained byKropfli and Miller (1976). The same kind of approachhas been used in hail trajectory studies at NSSL (Ziegleret al., 1983; Nelson, 1983) and at NCAR (Miller et al.,1983; Heymsfield, 1983; Miller and Fankhauser, 1983;Foote, 1984). Hail growth has also been studied usingthe growth trajectory technique within a model stormwind field (Xu, 1983). Other studies of precipitationformation within Doppler radar-derived wind fieldshave used more of an ensemble or Eulerian approachthan an individual or Lagrangian precipitation trajectory approach (e.g., Rutledge and Hobbs, 1983).Here, the approach is exclusively the single-particleone. The emphasis is necessarily on the larger particles,but not exclusively on hail. A total picture will requirehandling particle concentrations as well, but that, inthunderstorms, is a major problem in its own right.The field work was part of the Cooperative Convective Precipitation Experiment (CCOPE) in southeasternMontana in 198 1.2. Data reduction and qualityDoppler radar data were reduced and edited by interpolating raw reflectivity and radial velocity to aCartesian grid (0.8 km horizontal spacing and 0.5 kmvertical spacing) in a manner described by Miller et al.(1986) and Mohr et al. (1986). Additional subjective0 1986 American Meteorological Society10571058 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, No. 1 Iediting of the radial velocity data was done in stormregions where large reflectivity gradients existed, primarily along the upshear storm flank. Here, radial velocity estimates were particularly vulnerable to sidelobe contamination from reflectivity core regions.Horizontal motion was obtained by combining radial velocities from two or more radars and solving thedual Doppler equations in a least-squares sense. Vertical motion was derived by integration of the divergence field, assuming anelastic mass continuity, downward from storm top, defined as 500 m above the highest data level. The upper boundary condition is zerovertical velocity. A lower boundary condition of zerowas not applied here because of inadequate radar sampling within the lowest kiloqeter, due to the relativelylong radar-to-storm distances of 25 to 80 km. Comparison of vertical motion obtained by aircraft, froman objective interpolation scheme that has proved verysuccessful (Fankhauser et al., 1985), with the multipleDoppler analyses shows that unrealistically large updrafts appear within the primary updraft region at thelowest data levels of 1.8 km MSL or' 1.0 km AGL.Updraft contour shapes and locations from the aircraft- were quite similar to those from the radar analysis, but the radar updraft maxima were up to two times as strong as the aircraft maxima; at the time period most studied here, which happened to be the worst case, theaircraft maximum was 4.2 m s" and the radar, 1 1 m s-'. The lack of low-level data is not the cause of the errors, of course, but if low-level data had been available, use of a lower boundary condition as well as the upper one would have reduced them. Otherwise, the.greatest errors in u, o, w are thought to occur along the upshear (west-southwest) storm flank where the largest reflectivity gradients were observed. However, wind values over the entire storm volume appear to be sufficiently good to construct precipitation trajectories that are qualitatively correct. Our judgment is that errorsfrom the steady-state assumption are very probably more significant than those in the individual wind field,with the possible exception of the low-level updraft values.3. The stormOn 12 June 198 1, a convergence line oriented northsouth was present in the boundary layer and was visibleon radar as a distinct, weak line of clear-air echo. Itwas situated northwest of Miles City, Montana. Theline moved slowly eastward in the early afternoon, anda series of thunderstorms began breaking out alongit-the earliest some 150 km to the north, and laterones successively farther south. The southernmost vigorous convection started about 1530 LDT, about 50km to the north of Miles City, and grew into a smallbut vigorous storm that was both simple and rathersteady.in overall organization. This storm is the objectof study. (All times given hereafter are local daylighttime.)The storm slowly grew to peak intensity, which itmaintained from about 1630 to 1730, then slowly decreased in vigor and dissipated finally afabout 1830as it moved eastward at 9 m s-'. The environmentalsounding and the hodograph with storm motion aregiven in Fig. 1. Note the strong wind shear, the appreciable potential instability (5°C at 500 mb), and therelatively warm cloud base temperature, + 1 1 "C. Thehighest observed radar top is between 1 1 and 12 kmMSL, and the 0°C level is 4.1 km in the environmentand about 4.7 km within adiabatic cloud.The three-dimensional storm-relative wind field andthe radar echo structure at 1655, the time studied inthe most detail here, are shown in Fig. 2. The inflowis from the southeast and at low levels the storm hasa single, strong updraft several kilometers in diameterat its southwest end. With increasing height, the environmental wind quickly becomes strong from thewest-southwest, and the updraft loses its originalsoutheasterly winds in favor of the southwesterly windsof the environment. At the middle and upper levels,but not near cloud base, the updraft area extends morethan 10 kilometers to the northeast, and shows a morecellular character. Individual cells form aloft along theupshear flank and move to the northeast relative tothe storm itself, though in the time-lapse motion pictures taken from the University of North Dakota Citation 11, the southwest edge of the storm looks morelike a'continuous jet than a discrete succession of turrets. This is a matter of degree, of course. The University of Wyoming King Air did find small "turrets"to penetrate, but they rapidly developed a strong radarecho and ,merged with the bulk of the cloud, so thattypically it was not possible to make more than onepass through a turret.The trajectory modeling that follows uses a singlewind field as if the storm were steady state. It was not,of course, and it is a matter of judgment and possiblyof investigation in the future, how important that simplification really is. The upper-level cells shown in Fig.2 are unsteady over the time period of the trajectories,partly because they move at a different velocity fromthat of the storm. However, at altitudes of 7 f 1 kmMSL where they are most pronounced, the horizontalwind.velocity within them (see Fig. 2) carries particlesout into the anvil of the storm, from whence they cannot return. into the main updraft region. This unsteadiness is clearly not an important factor in' the trajectory ensembles described in this paper.Potentially more important is a cellularity that existsin the midlevels, but is not as marked. Figure 3 containstracings of the 45 dBZ, contour at 5 km MSL, aboutevery other volume scan, separated in the figure bygiving the storm an artificial, extra 1 km per minuteeastward velocity. One can discern here and in other1 JUNE 1986 CHARLES A. KNIGHT AND KEVIN R. KNUPP1059MILES CITY 1640200 - 12302520IY15w1050-5 0 5 IO 15 20 25 30 35 40 45u (ms")FIG. l(a). Thermodynamic diagram for the Miles City sounding, 1640 LDT, about 50 km south of the storm. Aircraft data confirm the1 1 g kg" water vapor-mixing ratio and the other cloud base properties. (b) Hodograph from the sounding of Fig. I, with cloud base indicated.Two of the storm motions used are indicated by x's-(u, u) = (8.9. 1.6) and (8.9, 3.6) m s"-within a dashed circle that has been added toemphasize the approximate nature of the storm motion (see text).data a new cell developing to the south of the old oneat 1649: 10 and taking over by 17 1550. This is not adominant feature of this storm, and we choose to ignoreit in the analysis; however, it remains as a qualificationin comparing the trajectory results with the radar echofield. .Doppler-analyzed ,updrafts exhibit horizontal scalesof 3-5 km and show maximum speeds of 15-20 m s-'near 5.3 km MSL during the 1630-1730 time period.Because scales of 3 km and less are severely attenuatedby the Doppler'analysis, actual updraft speeds probablyexceed 25 m s-' on small scales. Downdrafts of 5 ms-' maximum amplitude are most common within theflank just downshear from the primary updraft in thelowest 4 km.There is usually a fairly abrupt radar echo gradientand an echo overhang to the south at the southern sideof the updraft, and there is a suggestion of a hook echoto the south at 2.8 km (Fig. 2). This feature was bestdeveloped before 1655 and at lower elevations, and isin the process of disappearing at the time of Fig. 2.The maximum, 10 cm-equivalent radar reflectivityfactor of this storm reaches 70 dBZ, several times atlow levels, but the high values never extend very highin the atmosphere: 60 dBZ, reaches about 6 km MSLbriefly, and 50 dBZ, reaches about 8 km. A more detailed description of storm characteristics will appearin a subsequent publication.4. Trajectory modeling proceduresa. General philosophyThroughout this study an effort has been made tokeep the computations as simple as possible. The microphysical specifications for the particle growth equations have been simplified to an extreme degree; doingthis has several justifications and benefits in a study ofthis type. First, the Doppler-derived wind field is bothhighly smoothed and somewhat fallible. No purpose isserved by carrying the precision of the microphysicalpart of the computation past the point where the totalerror is dominated by that in the wind field specification. While there is no way to tell just where this pointlies, our feeling is that it probably lies on the side ofmicrophysical simplicity. Second, the only test for correctness of the deduced trajectories is the radar echopattern itself. Since we can look only at individual particle growth trajectories and do not treat particle concentrations at all, this is fundamentally a rather roughtest. The measured reflectivity factor 2 does dependimportantly upon concentration n in addition to particle diameter D (2 = En@), especially since it is moreor less proportional to dD. There is little point in adding complexities to the procedure beyond the level ofsensitivity of the verification. The sensitivity tests willbe a test of the "validity" (in a practical sense, of course)1060 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, No. I1-Itn2EaqYNLd BZ w m s" (u. v)FIG. 2. Radar echo, updraft and the horizontal wind field shown at 2.8, 4.8 and 6.8 km MSL (2,4 and 6 km AGL), 1655 LDTscan. Note the rapid decrease of the maximum equivalent reflectivity factor with height and the downwind extensions of the updraftwith height. The horizontal winds are storm-relative, using a storm speed (u, u) = (8.9, 3.6) m s". In the vertical velocity plots, thecontour between the blank and cross-hatched regions is 0 m s". The 2.8 km level is a little above cloud base.of the simplifications. Third, these calculations alwaysproduce a deluge of numbers, and the simpler the wholeframework is, the easier the computations are to checkby hand, and the easier they are to assimilate intuitively.Thus, to start, we want to err on the side of being toosimple and add complexities only when they seem tobe both needed and justifiable in terms of the availabletest and the overall purpose.b. Growth specificationsCollection efficiencies are unity, so all liquid watercontents are actually eflective liquid water contents.Particle density is a function of size only; it is completely independent of growth conditions. Two densitycases are distinguished solid ice, and a linear changefrom 0.5 g cm-3 at R = 1 mm to 0.7 at R = 1 cm:p = 0.915orp = 0.21 1R + 0.489,where p is in g ~m-~, and the radius R (cm), has animposed upper limit of 1 cm in this study. Growth' isby geometrical sweep-out, all particles aie spherical,and terminal velocities are calculated using the dragcoefficient (CD) formation. Three cases are used:Case 1: p = 0.915, CD = 1;Case 2: p = 0.21 1R + 0.489, CD = 0.75; andCase 3: p = 0.21 1R + 0.489, CD = 1.5.Liquid water contents within cloud are based upon theadiabatic values (LWCA) as follows. To simulate entrainment, we consider the liquid water content to beadiabatic for updraft speeds w > 10 m s", and to decrease linearly at constant altitude from L WCA at w= 10 m s" to 0 at 'w G 0. We further decrease theliquid water content linearly from the value as deter1 JUNE 1986 CHARLES A. KNIGHT AND KEVIN R. KNUPP 106 145 dBZ Contours Iat 5km MSL1649 : 101642:45 ?1 1 I I I I I I-15 -5 5 15 25 35 45 55 Km EAST OF RADAR AT 1636 :30FIG. 3. The 45 dBZ, contours from 5 km MSL CAPPIs are shown for a period embracing the1655 LDT orimarv analysis time. Generation of new cells to the south does occur but it is not a"dominating feature.mined above at the -20°C level, to 0 at the -40°Clevel to approximate the effects of glaciation and depletion. Note that the density and terminal velocityspecifications in this work are much simpler than thoseused in most other precipitation trajectory studies, butthe liquid water scheme is comparable.The densities are high compared to typical HighPlains graupel, as reported in Heymsfield (1978). Thiswas done on purpose, with support from two pieces ofdata. First, the cloud base temperature was unusuallywarm for the High Plains (+ 1 1 "C), which would tendto produce unusually high liquid water contents-otherthings being equal, of course-and therefore high density. Second, the few two-dimensional (2-D) images(from the PMS 2-D probe) of precipitation withincloud, gathered by the Wyoming King Air at about-10°C (5.5 km MSL) at the extreme southwestern edgeof the radar echo, showed only very round images (todiameters of 1 mm and a little more), suggesting highdensity particles-perhaps even frozen drops.Growth is allowed under the specifications anywherebetween cloud base and the -40°C level. The fact thatice does not grow at temperatures above 0°C is, for themoment, disregarded. Sensitivity to this simplificationis tested later.The relation beween terminal velocity and time determines a particle's growth trajectory within a givenwind field. The growth specifications used in this study(cases 1,2 and 3 above) have the result that at constantliquid water content and pressure, the terminal velocityis nearly linear with time (pointed out by Browning,1963), right from zero size. The slope of this line is afunction of CD, independent of p. Real processes, ofcourse, often introduce large deviations from this atthe small sizes. The onset of liquid coalescence andthe vapor growth and early riming of snow particlesintroduce delays of up to 10 minutes or even more inthe very early growth stages, and new particle formationby shedding or break-up would bypass the smallest sizesentirely. Implications of this unreality at small sizesare discussed at the end of the next subsection.c. Computational proceduresThe time step used was 10 seconds; a test with aone-second step showed little difference.The only departure from previous work was to runthe calculations backward, starting with arrays of precipitation particles of final, specified sizes at 2.8 kmMSL, about 400 m above cloud base, and moving thembackward in time through the wind field while decreasing their mass and size according to the growth equation and their density. Particles were started on a l kmgrid within a rectangle enclosing the region of substantial radar echo. Particle diameters (Do) at the level atwhich the computations started were typically 0.5, 1 .O,I .5 and 2.0 cm, spanning the sizes of interest in a stormof this size and strength.Discussing the backward calculations of the trajectories in an intelligible way has proved difficult. Thetrajectories themselves will usually be discussed in thesense of real time, not in the backward sense of thecalculation. At the beginning of a trajectory, the particleis smallest. At the beginning of a calculation, the particle is largest. When the calculation is discussed, thebackward time sense will be indicated explicitly or withquotes, as in the following sentence. Suppose we "start"a particle with Do = 2 cm at location (xo, yo, ZO) andrun the trajectory and the growth equation backward.The computed trajectory might be a realistic trajectoryin the storm or it might not; correspondingly, a particleof 2 cm diameter might have existed at (xo, yo, zo) orit might not. The method is to test the trajectories forreasonableness, reject those that seem unreasonable,and then compare the field of (xo, yo, zo) locations thatlead to reasonable trajectories, with the radar echo pattern at ZO. In this study zo is always a single value, 2.8km MSL, about 400 m above cloud base. This reducesproblems with melting, as discussed below.1062JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, NO. 11One of two things can happen to a particle in thecomputation: it may shrink to zero or it may leave thedata field. (In practice, a third possibility was for theparticle to survive for 40 minutes, at which time thecomputation stopped automatically. In these cases, aparticle after 40 minutes of computation was alwayssmall, and its trajectory was always accepted as reasonable.) Two types of criteria are used to judge trajectories unreasonable and reject them. One is basedupon a comparison of particle size with measured radarreflectivity and the other upon the conditions when aparticle leaves the field of (u, 0, w) and/or dBZ,. Therejection criteria were designed to be generous, so asto be sure to err on the side of accepting too manytrajectories rather than rejecting too many. This is inkeeping with the general philosophy emphasizing thefallibility of the wind field itself and the simplifiedgrowth specifications.The first two rejection criteria simply eliminate trajectories if at any point a particle is too large for thevalue of the radar reflectivity factor at that point:1) Reject if, at any point, dBZ, d 10 and D 3 0.52) Reject if, at any point, dBZ, d 30 and D 3 1.0These were instituted at the very first and kept unchanged with one qualification noted below. Whilesingle paiticles in the storm might violate one of thesecriteria in fact, it seems certain that a significant concentration would not, except perhaps for large hail, forwhich very small concentrations are significant in somecontexts. The equivalent radar reflectivity factor for0.1 m-3 of particles with D = 0.5 cm is 18 or 14 dBZ,,with the two density -specifications, and that for 0.1m-3, of particles with D = 1.0 cm is 37 or 33 dBZ,.This is without any other sizes `being present at all.There were some steep reflectivity gradients (unfiltered values up to 40 dBZ, km-') in the low levels ofthe 12 June storm, so a modification to criteria 1 and2 was made:3) Criteria 1 and 2 do not apply for the first fiveminutes of each (backward) trajectory calculation.This is because we didn't want to reject a trajectorybecause the particle reached cloud base only a shortdistance away from a high reflectivity region. In factthis addition made very little difference in the results,but it seems, in general, to be a physically reasonablerelaxation of the first two.Two final specific rejection criteria were institutedto eliminate most members of two classes of obviouslyimpossible trajectories in this storm that were othenvisebeing accepted. Both refer to the conditions when aparticle leaves the data field in the computation (entersthe data field in reality; we refer to this point as thestart of a trajectory). At the bottom of the field, as notedcm;- cm.above, the updrafts were unrealistically high in someplaces. This allowed particles several millimeters in diameter to start at the bottom and rise into the mainupdraft, starting to grow when they p+ cloud baselevel. It seemed unreasonable to allow such large particles to originate so low, and a fourth rejection criterionwas instituted. In effect, this is a rough compensationfor the fact that the low-level updrafts are too strong.4) Reject if particle originates at 1.8 km (the lowestThe fifth and final rejection criterion involves particles that start at the top of the radar data domain.The reflectivity rejection criteria still allowed particleswith D up to 0.5 cm to originate at the top edge of thefield and at the sloping, western end (upwind) of thestorm. Since there was no upwind source, and obviously no way for particles that big to get there, thefinal rejection criterion was designed to eliminate thepossibility of embryos (0.2 cm < D d 0.5 cm) originating at the top and upwind sides of the field.5) Reject if particle originates at the edge of the fieldwith D 3 0.2 cm and TV- w 2 5 m s-',where TV is the terminal velocity; therefore, TV - wis the earth-relative, vertical motion of the particle; thiseliminates sizable particles falling into the storm at adownward speed exceeding 5 m s-'.It is worth reiterating that these criteria are very generous; the purpose is only to eliminate the most unlikelytrajectories automatically and objectively, so as to reduce the amount of data that requires mental assimilation. The last three were instituted in response to theactual data from runs with just the first two, but therewas no tuning of the criteria in response to runs usingthem. They have all remained unchanged.The procedure of calculating the trajectories backward has the primary advantage that the array of starting points that needs to be examined is two-dimensional rather than three;' and to the extent that theinterest is concentrated on hail, the two-dimensionalarray need only cover the region of fairly strong radarecho. The embryo source re&ons in this and otherstorms tend to be more extended. Furthermore, themajor uncertainties in all the growth specifications areat the smallest sizes, not the largest, so doing it thisway puts them at the end of the calculation rather thanat the beginning, where they can influence the wholecourse of the trajectory. (A ,reviewer pointed out, corlevel) with D 2 0.1 cm.' This is true when a steady state approximation is employed,it is here. The advantage of the backward scheme must be far greaterwhen evolving wind fields are u.d, if the "ground truth" is essentiallyinstantaneous: radar data or a shortduration hail collection. In thatcase the starting arrays are two-dimensiond rather than four, withtime yet another variable for starting embryos in a forward cdculation.If the ground truth is time-integrated, then either a forward or back.ward calculation has to involve time.1 JUNE 1986 CHARLES A. KNIGHT AND KEVIN R. KNUPP 1063rectly, that the updrafts are especially fallible in the lowlevels; this can be used as a counterargument.) Ofcourse, if one starts with large embryos in a forwardcalculation, as has been done in most hail trajectorystudies, this factor makes very little difference. Wethink, however, that treating embryo size as an independent variable, which this procedure encourages, canbe a very misleading thing to do. Embryo size and location are not independent because the same growingparticle has a range of sizes and locations.*It turns out, as might have been expected, that inone respect the forward and backward calculations aresimilar. The location and/or size at the end of a trajectory calculation can be very sensitive to the locationand size of the particle that is put in at the start of thecalculation, whether that particle is the final hailstoneor the starting embryo. Each direction of calculationhas advantages for some purpose; but if the object isto determine where the storm might provide hail ofgiven sizes at a low altitude, the regions of origin ofthe hailstones, and what the relative importance of different types of trajectories is, then the backward schemeseems far more economical.Because of the increasing errors at the smallest sizes,the trajectory calculations to be presented are invalidnear their ends. They can only be viewed as suggestinggeneral vicinities of actual origin. Either a more detailedgrowth model or a qualitative judgment must be substituted for the present scheme when a particle reachessome small size. This will be discussed a little morefully in a later section.5. Resultsa. Sensitivity testsSignificant sensitivity of the trajectories to the prescribed storm motion and to all of the microphysicalspecifications was expected, and the first step has beento try to rank these in terms of importance. Storm motion enters in because the velocities that are measuredare relative to the earth, whereas the important velocities for the trajectories are those relative to the storm.This was a concern both because storm motion issomewhat arbitrary to measure and because it often isnot clear what time or space scale to use to estimateIn fact, a major advantage of the backward trajectory schemeover the forward scheme employed in other studies is that it avoidsentirely the concept of embryo size. It is difficult to imagine a useful,objective criterion to decide when a particle stops being an embryoand starts growing as a graupel, or hailstone, or some other categoryof particle. Hailstones themselves, especially the smaller ones, don'tprovide any single, natural demarcation (Knight and Knight, 1970". . . the determination of where the embryo stops and the hailstoneproper begins is truly arbitrary . . ."), and their calculated trajectoriesoften don't either. Calculating the trajectories backward puts the sizevariable where it belongs and can be tested more easily-with theend product-and avoids the whole concept of embryo size.storm motion. The concept of storm motion is linkedwith the steady state approximation, in that the motionof a perfectly steady storm would be perfectly welldefined and easily determined. Storm motion becomesa problem to the extent that the steady state assumptionbreaks down.At the very beginning an overall storm motion wasdetermined, averaged over more than an hour intervalbracketing the 1655 scan being used. The trajectorycalculations were performed for that overall stormmotion (u, v) = (8.9, 1.6) m s", and then four moresets were calculated for u f 2 and o f 2 m s". Theresults, for Do = 1.0 cm and case 1 (solid ice, CD = 1)are shown in Fig. 4. Here the prescribed trajectory endpoints are on a l-km grid. Each "A" represents theend point of an accepted trajectory: a particle with diameter of 1 .O cm that could have grown and ended upat that location at cloud base according to the growthspecifications and the rejection criteria. Both the shapeand the location of the area of accepted trajectories isfairly sensitive to the storm motion.While the decision is not clear-cut, we chose the bestfit with the radar echo pattern to be (u, v) = (8.9, 3.6)m s-I, both for having accepted particles in the strongestecho region and for some indication of the "hook" tothe southwest. It is not feasible to present all of thecombinations of storm velocity, microphysical specifications and starting diameters that were run, but astorm motion of (8.9,3.6) m s-' looked the best, overall,and that is the value used for the rest of the study.However, because this motion was different from theoverall storm motion measured at the start, it was examined in a little more detail. Since the maximumradar echo is automatically plotted on all of the PPIs,a simple, objective scheme is to plot that location, atcloud base, for each scan. This is done in Fig. 5 withfour of the times labeled. (The times are not all consecutive along the track from west to east. Indeed, themaximum echo need not always have a continuoustrack, so the positions are not connected in the figure.)It is interesting that the storm motion up to about 1655does turn out to be closer to (8.9, 3.6) than (8.9, 1.6);but the main point of the figure is the difficulty of defining a storm motion, especially after 1655. As sensitive as the trajectories are to the storm motion, thisseems to be an important caution and qualification inwork of this kind.3A comparison of the three cases of density and dragcoefficient specifications is given for DO = 1.0 cm inWhen the storm motion is very difficult to define, the implicationmay be that the steady state assumption should not be used. However,in studies that employ a time series of wind fields rather than assumingsteady state, the usual procedure is to use a storm motion to advectall the fields to the same location. One expects an appreciable sensitivity to storm motion here, too; sometimes, perhaps, even morethan with the steady state approximation, though no one has testedit, to our knowledge.1064 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, NO. 11I Case I(8.9, 1.6140, 50, and 60 dB2 ContoursI Case I (8.9, -0.4)ICase I (8.9, 3.6)II*I ICase I(6.9, 1.6)1655 o0 =i.OcmFIG. 4. The locations at 2.8 km MSL of particles 1.0 cm in diameter, grown in the wind fieldusing microphysics case 1 (see text), for the five storm motions including and bracketing (u, 0)= (8.9, 1.6) m s". The 40, 50, and 60 dBZ, contours are shown in each case, and the scale isindicated by the I-km spacing of the A's. Only the 1-km grid points on and to the south of thehorizontal line were tested. At grid points not marked with A, growth trajectories leading-to 1 .Ocm particles were rejected-judged impossible-according to one of the rejection criteria in thetext. The patterns and locations of the A's are fairly sensitive to storm motion. Their correspondencewith the high radar reflectivity is a test of the scheme.Fig. 6, the top row. They are all fairly good, in termsof matching the reflectivity pattern: cases 1 and 2 areperhaps somewhat better looking than case `3. Again,it is not reasonable to present the results for all Do (oneexample including a range of Do will be given later),but we choose case 2 for most further presentations.In general, the choices here are more difficult, and itseems clear that the overall results are somewhat lesssensitive to these microphysical factors than to thestorm motion, in the ranges of both factors that aretested.Figure 6 also shows the same three cases run withone-half the original liquid water content specification.Obviously the sensitivity to liquid water content is verystrong, more than that to storm motion or density anddrag coefficient. (Note that the real uncertainties in thedrag coefficient are a good deal less than the factor of2 between cases 2 and 3, while the uncertainty in liquidwater content is about that tested, or even greater;however, the "definability" of storm motion is aboutthat tested, or even greater in some instances.)The next sensitivity test was to estimate the impor1 JUNE 1986TNCHARLES A. KNIGHT AND KEVIN R. KNUPPII I I IIx : Z max. 2.0 to 2.5 km MSL(9. I .4.2) m s"xx xX' 1712Overall (8.9,1.6) ms"?(l0.6,6. I) m s2 A 1730 .1065FIG. 5. The locations of the maximum radar reflectivity factor at 2.0-2.5 km MSL plotted foreach volume scan between about 1610 and 1730. The overall (u, u) for the storm was estimatedto be (8.9, 1.6) m s", but the motion for more limited periods of time could be rather different,as indicated. The lengths of the arrows are not significant.tance of the freezing level. One step toward realism is and again, this judgement is supported by examiningto prohibit growth below the freezing level at diameters the other values of Do. In terms of the purpose of thisgreater than 5 mm, but still without including the pro- study, we conclude that it is not important to includecesses of melting and shedding. The result of this mod- a specific treatment of melting, shedding, drop breakification is shown in Fig. 7. There is very little effect, up and the like, even with the approximately 2 kmICose I (8.9, 3.6)IACose 2(8.9, 3.6) I I Cose 3(8.9, 3.6)II1655 Do = 1.0 cmFIG. 6. With the same presentation used in Fig. 3, the three cases of density and drag coefficient (case 1, solid ice, C, = 1.0; case2, lower density, CD = 0.75; case 3, lower density, C, = 1.5) are compared in the top row, for a storm motion of (8.9, 3.6) m s",and in the bottom row with one-half the liquid water content. Sensitivity to the density and drag coefficient is relatively low, but toliquid water content very high.1066JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, No. 11I Case 2(8.9, 3.6)I I1655Do= 1.0 cmFIG. 7. Case 2, storm motion (8.9, 3.6) m s" was run withoutallowing growth at diameters larger than 0.5 cm between the cloudbase and the melting level. Comparison with the same case in Fig. 5suggests that ignoring the melting level does not have a very importantinfluence on the.ensemble of trajectories.separation between cloud base and the freezing level.If better data for testing the results were available, suchas precipitation size spectra at the zero degree level,then including liquid-phase processes might very wellbe desirable.Before summarizing the results of the sensitivitytests, it may be useful to discuss the context again,because it is different from that of some other trajectorystudies. Sensitivity here does not mean sensitivity ofan individual trajectory, calculated either forward orbackward from any initial size and location. Examplesof great sensitivity to each of the specifications can befound, looking at individual trajectories. Here theviewpoint is that of the ensemble ofpossible trajectoriesleading to given-sized particles at cloud base, certainlya more meaningful measure of sensitivity overall.Bearing this distinction in mind, probably the majorresult of these sensitivity tests is that, at least in thisstorm but perhaps also in general, by far the most important and uncertain specification in doing trajectorystudies is that of liquid water content. Next most important may well be storm motion. There seems to belittle point in putting a great deal of effort into improving either the density or the terminal velocity formulation over the extreme simplicity of the presenttreatment, without a better liquid water content formulation and better data to test results. It is noteworthythat a similar, dominating sensitivity of particle growthto liquid water content, over any other microphysicalfactors, has been shown by King (1984) for simplestratiform clouds.It is also noteworthy that our results on sensitivityto liquid water content contrast with those of Nelson(1983) and Ziegler et al. (1983). There are several reasons for this. One is that they keep account of hailstonetemperature, and their formulation assumes sheddingof all excess (unfrozen) water when the hailstone temperature is 0°C. Our formulation in effect assumes noshedding at all. Obviously when a hailstone is at 0°Cin their formulation, as it often is in their cases, thegrowth is strictly independent of changes in the liquidwater content. Reality lies in between the two simplifying assumptions, since shedding does depend uponair temperature and hailstone size (Carras and Macklin,1973). Another reason for the different sensitivity toliquid water content is probably the style of presenta-.tion: the fact that the test here is the size and locationof the area of accepted trajectory end-points near cloudbase, while Nelson (1983) and Ziegler et al. (1983)looked at individual trajectories 'in detail. The sensitivity result reported here is to be expected since hailsize must be sensitive to liquid water ,content if onedoes allow spongy growth and if the residence time inthe updraft is largely controlled by advection acrossthe updraft rather than by falling through the updraft,as it is in the 12 June 198 1 storm.The rejection criteria have no influence on the trajectories themselves. Since the results turned out ratherwell, there seemed little point in testing sensitivity tothese criteria. In looking at trajectory types' in detail,however, it is useful to examine some of the rejectedones also, to see why and where they were rejected.b. TurbulenceAs a final test, it would be desirable to determinethe importance of the smoothing of the actual threedimensional turbulent wind field (which is not measurable) as a consequence of Doppler analysis procedures. We do not know how to do this realistically.Convective cloud turbulence levels and scales are notwell known, and certainly vary spatially and temporallywithin a storm (e.g., Knupp and Cotton, 1982). Furthermore, with the present scheme of computing thetrajectories backward, it is not completely clear whatthe results mean. Adding random fluctuations to thevelocity field and running the program several timesfor a given a, yo, 20, Do does give a spread of calculat$particle starting points and results in a percentage ofaccepted trajectories rather than a simple yes or nodecision, but it is not easy to interpret in terms of, say,particle concentration, because other important factorsare involved. For instance, Young (1978) calculatedthe influence of turbulence on particle size distributionin a simple model. Nevertheless, this procedure doesgive the acceptance fields an added realism, and it was 'of interest to see if it would change any of the resultsdrastically.Fluctuations were added to the u, v and w components of the velocity by adding several sin functionswith random coefficients. All fluctuations were a func1 JUNE 1986 CHARLES A. KNIGHT AND KEVIN R. KNUPP 1067tion of time only, and were applied along each particletrack, at each time step. The function used was6u', u`, and w' = K 2 sin 27r - + Ri ,i= 1 li 1where t is the time in seconds, K is a constant determining the maximum possible amplitude of a fluctuation (if X = 2 m s", the maximum possible fluctuationis 12 m s-'), Ci varies over the six prime numbers between 5 and 19 inclusive, and each Ri is a randomnumber between zero and unity, selected anew for eachtrajectory. Time equals zero at the start of each trajeotory computation where D = Do. This function waschosen to have no effective periods longer than about20 seconds, and, in retrospect, the terms with the lowestthree or four C values surely make no difference, sincethe time step of the computation is 10 s. The resultsare shown in Fig. 8 for case 2, (u storm, u storm) = (8.9,3.6) m s-I, K = 2, and without the growth restrictionat temperatures above 0°C that was included in Fig.7. (The fluctuation study is quite expensive in computertime, and it did not seem worthwhile to redo it to include this factor.) The numbers in the figure are thenumber of accepted trajectories out of 10 runs at eachlocation.This technique does show features that might notbe seen as clearly otherwise, The "hook echo" at thesouthwest end of the storm, foc instance, has only relatively unlikely trajectories in its vicinity for all DO.Also, the results at DO = 2.0 cm mostly show relativelyunlikely trajectories. This kind of.scheme could beuseful for studying the origin of a storm's largest hail,but it does not .reveal much that is new in the presentcase.c. The trajectoriesWithin all limitations and approximations, andconsidering the rough nature of the check using theradar echo pattern, these results look very good. Wewould not have expected this good a fit with so littletuning of the specifications. One further test for trustin the results is to look at other times at which the echopattern at cloud base was distinctly different and seethe extent to which the differences between the trajectory results mimic those between the echo patterns.CASE 2 (8.9, 3.6) 1655Do = 0.5 cmDo = 1.0 cmDo = 1.5 cm Do = 2.0 cmFIG; 8. Ten runs made at each location, using the velocity fluctuation scheme described in thetext. The number of accepted trajectories out of the ten is plotted at each point (zeros omitted).1068JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, No. I II637CASE 2 (8.9,I7203.6) Do=I.O cmFIG. 9. Two other scan times tested by using exactly the samespecifications as those found best for 1655 LDT. The changes in thepatterns of accepted trajectories (Do = 1 cm) are similar to the changesin echo patterns. The east-west line at 1637 LDT is the southernextent of the tested Do's at that time; the north-south line at 1720LDT is the eastern extent at that time.Figure 9 shows the results for 1637 and. 1720, maintaining the same conditions-case 2, storm motion(8.9, 3.6) m s"-as were used for 1655, and using thesame Do = 1 .O cm for the illustration. The comparable1655 case is the center top panel, Fig. 6. The tendenciesare reproduced fairly well. (The displacement of the, pattern of accepted trajectories from the area of stron gest echo at 1720 is not surprising in view of the prob able error in storm velocity suggested by Fig. 5.) With this degree of agreement between the trajectories and the radar echo, we feel justified in discussing the cal culated trajectories as if they are qualitatively correct.There is an immense amount of information in thesets of calculated trajectories, and it would be difficultand probably pointless to try to communicate most ofit in detail. In the remainder of this section we illustratesome of the variety of trajectory types and startingpoints, for the variety is important in itself. Then thefollowing three subsections discuss briefly three particIt becomes easier to classify separate trajectory typesas Do increases, because the number of accepted trajectories decreases. Figure 10 shows the accepted cloudbase trajectory end points at De = 0.5 cm, along withthe three Cartesian projections of the starting point location^.^ The actual particles start throughout thecloud, ending up with Do = 0.5 cm over much of theregion of cloud base with strong radar echo. Of coursethere are correlations between starting regions andending regions, in fact, but the results are complicated.Perhaps the most interesting aspect of Fig. 10 is the, ular aspects of the results that seem interesting.Recall that a starting point is either the point at which D goes tozero in the backward calculation, or where the particle leaves thefield of data, at which point its diameter must be less than 0.2 cm tohave a chance of avoiding rejection. In fact, of the 35 trajectoriesaccepted for the case shown in Fig. 11, 21 went to zero size and 14left the field of data; and of the 14, only two had D > 0.1 cm whenthey left.fact that acceptable trajectories do not produce particleswith D - 0.5 cm over a significant portion of the areaof strong radar echo. (The contour in the top panel is40 dBZ,.) At the southern side of the storm, the southeasterly flow near 2.8 km (Fig. 2) is such that there isinsufficient time to get particles of this size. The rejection criterion that operates for most of these is number5): the particle enters the wind field with TI, - w 3 5m s" and D 3 0.2 cm. (This also usually occurs atdBZ, values of zero or below, so the rejection wouldseem well-justified on that score also.) The southernecho boundary is more-or-less vertical, and the entryof these would have to be into this vertical wall fromthe southeast at diameters between 0.2 and 0.5 cm.The blank area within the 40 dBZ, contour to the north(top panel, Fig. 1 1) is caused by the downdraft in thatregion (see Fig. 2). The particles would have had toenter the top of the storm at too big a size, and rejectioncriterion number 5 is again the cause.Both of the areas within the 40 dBZ, contour thatjack accepted trajectories with Do = 0.5 cm may beartifacts of the steady state assumption. In connectionwith Fig. 3, it was noted that the northern part of theEx>CASE 2. I I(8.9,3.6)Do=0.5cmI655AAAAAA' A nannn-aJI AAAAA'AAAAAAWn'A AA AAAbJAAAAAAA/ AAA5247..L10.8 7.8 4.8 1.8-8.0 -3.0 2.0 7.0 12.0X (km)FIG. 10. The accepted trajectory end points for 0.5 cm diameterat cloud base are shown at the top. Beneath are the three projectionsof the field of corresponding starting points.1 JUNE 1986CHARLES A. KNIGHT AND KEVIN R. KNUPP10691655reStag.PointInflowSP:N:2- 7S4-7 km' 3.5 km4.3-5.3 km3.4-5.2km A A A A A A A A F.i-G,Jkl 50dBZ465.5-6.3 km24 dBZ2.6-5.1 kmOdBZIpanels show the accepted locations near cloud base., with a classification of trajectory types (left)and examples of each, projected onto the horizontal, with starting and maximum heights anddBZ, at the start. The bottom panels show starting locations and altitude ranges (left) and the 2.3km radar echo pattern (right).storm at 1655 is a. dying cell, while the southern partis growing. Thus the downdraft that prevents acceptance ofDo = 0.5 cm trajectories to the north may havebeen a recent development at 1 6 5 5, as well as the stronginflow to the south that prevents acceptance there.Figure 1 1 illustrates aspects of the trajectories for Do= 1 cm. At the upper left, the accepted DO cloud baselocations are classified into five named trajectory types,and at the upper right, (x, y) projections of typical,individual trajectories are plotted, labeled with thestarting altitude, the high point of the trajectory, andthe value of the radar echo intensity at the startingpoint. At the lower left, an (x, y) projection of all thestarting points is labeled with initials of the trajectorytype names and the altitude range of the starting pointsin each group, and at the lower right the same plot ofstarting points is superimposed over some radar echocontours at cloud base.The five types are quite simple, though one Do location is i'ncluded in two types-it starts as inflow andends as stagnation point-while one other does not fitcomfortably in any. Particles following a stagnationpoint trajectory originate, grow, and fall out close tothe flow's stagnation point at the upshear end of thestorm. Most of them originate within appreciable radarecho, then rise and fall. The inflow trajectories originatenear cloud base on the south, mostly at very low echointensity, move into the storm at low levels in thesoutheasterly inflow, then rise and fall in associationwith the main updraft and the echo core, which aremore-or-less colocated (Fig. 2). The N side trajectoriesstart at a variety of levels in and to the north of theecho core, and grow in what looks like an eddy in thehorizontal flow that produces very weak horizontal velocities on the N side of the updraft core. The S sidetrajectories start at a wide range of elevations south ofthe stagnation point and extend around the south sideof the main updraft. The core trajectories start at middle levels within the reflectivity and updraft core, riseand grow in the middle- to upper-level extension ofthe updraft to the east-northeast, illustrated in Fig. 2.It is gratifying to note that the highest point on all ofthese trajectories is at about 7 km MSL (about - 15 to-20°C). This fits with the vertical radar echo structureof the storm noted previously, and is obviously due tothe small and relatively weak main updraft and thefact that above 7 km the horizontal storm-relative windwithin the updraft becomes quite strong (Fig. 2).It is noteworthy that at Do = 2.0 cm, with only sixaccepted Do locations, there are one stagnation point,two inflow, and three core trajectories.We exercise some skepticism about the stagnation1070 JOURNAL OF THE ATMOSPHERIC SCIENCES' VOL. 43, No. 11Ipoint trajectories for this storm-though they are certainly reasonable in principle-because they lie in aregion where the wind field is especially fallible becauseof strong gradients, as mentioned in section 2; andabout the core trajectories because depletion of liquidwater in the region of very intense echo is probablygreater than that which the liquid water specificationproduces. The N side trajectories might also be influenced more by depletion than has been estimated, butare perhaps less likely to be seriously affected than thecore trajectories. The inflow and the S side trajectories. look the most trustworthy both on the basis of the dataand because they are the ones that, at Do = 1 .O cm andlarger, result in particles within the very strongest radarecho region at cloud base (compare the upper-left panelof Fig. 11 with the top center panel of Fig. 6).Overall, the results are roughly consistent with theecho pattern, and the analysis really only justifies theclaim that all of the .qualitative trajectory types areprobably represented within this storm, to some degree,and, that the inflow trajectories and the S side ones maybe the most important.The variety and extent of the source regions for the1 cm and larger particles, in a storm of the structuralsimplicity of this one, seems worth stressing. It does.not bode well either for hail suppression via the idea. of supplying more embryos to compete with the naturalones or for eventual parameterization of the wholeprecipitation process in any simple way. However,proper evaluation of these problems is not possiblewithout including particle concentrations in the analysis, as well as complete size distributions.d. The hook echoThe hook echo was a rather short-lived feature, thatis seen in a smoothed form both at 1655 (Fig. 2) and1637 (Fig. 9), but not at 1720 (Fig. 9). It was mostpronounced about halfway between 1637 and 1655, ataltitudes at and below cloud base, and is shown in Fig.12. While the feature is not reproduced well in themaps of accepted trajectories, those trajectories thatproduced particles most nearly in the hook echo suggestthat it was not caused by the environmental windssweeping precipitation-sized particles around the southside of the storm. Rather, for the brief period when thehook was present, the near-surface and cloud-basewinds relative to the storm at its location were almostdue easterly, and the winds at 5-7 km,were very lightor almost due westerly. This provided a variety of lowand midlevel trajectories by which particles could growand fall out in the hook region without being transported northward into the main body of the storm.These wind features can be seen in Fig. 2. As the hooklike feature disappeared, all of the low-level windsslowly changed, gaining an appreciable southerly component relative to the storm. This differential flow is adifferent means of producing a hook echo (or a hooklike echo) from the more highly rotational flow that1643~46 ' 1.4O X: 62 dBZI5 kmFIG. 12. The hook echo at its maximum development, seen in a 1.4" PPI near cloud base altitude.seems to be established for some tornadic storins(Brandes, 1981).e. Particle originsAs was noted in section 4, an advantage of doingthe trajectories backward is that the greatest uncertainties are at their ends, in the sense of the computation, rather than their beginnings. Thus the presenttreatment can be used, within limits, to determine general regions of origin for particles that can reach particular sizes at particular locations at cloud base, without detailed attention being paid to the microphysicalmechanisms of origin. The general region of origin canthen be used to limit the possible mechanisms of origin.Referring back to Fig. 1 1, for instance, it is clear thatsome of the 1 cm particles that were classified as resulting from inflow trajectories must originate as rainor drizzle in the inflow near cloud base, since the freezing level is about 4 km. In fact, the Queen Air 306Daircraft did encounter rain and drizzle in just theselocations in the storm inflow below cloud base. Liquidcoalescence processes cannot be ruled out as a potentialorigin of the liquid drops. (Recall the + 1 1 "C cloudbase temperature.) Since several of these trajectoriesenter the high reflectivity region before rising abovethe freezing level, shedding or melting processes couldhave originated some of these particles as well. A different type of study, with a much more detailed microphysical treatment, is needed to study particle origins in detail (e.g., Heymsfield, 1983). The present kindof treatment can limit the possibilities, and can suggesthailstone embryo types. In this storm, many hailstonesfollowing the inflow trajectories would almost certainlyhave drop embryos; and those following the other typesof trajectories, would be expected to be mixed, sincein all cases the altitude range of origin brackets thefreezing level.1 JUNE 1986 CHARLES A. KNIGHT AND KEVIN. R. KNUPP 107 1Many trajectories originate at zero size within strongradar echo, suggesting a possible important role forsecondary processes. In any case, the data that do existfrom the high-reflectivity regions of high-plains thunderstorms show that there is usually a more-or-less exponential particle size distribution, with an abundanceof small ice particles when the radar echo is intense(Knight et al., 1982). Strong radar echo from large particles alone does exist occasionally, but is fairly rare.J: Residence timesTable 1 shows the average residence times of particlesfollowing the accepted trajectories in the storm as afunction of final size Do, both with and without thefluctuations superimposed upon u, o and w described. earlier. It also gives the number of trajectories that exceeded the 40-m time limit in the program. It is inter' esting (and counter to our initial expectations), that,on the average, the smaller particles remain within thestorm the longest. After examining the trajectories withthe long residence times, however, the explanationquickly became clear. To grow large, a particle has tobe supported in a strong updraft, where high liquidwater contents occur and the growth rate is rapid. Sucha particle either leaves the updraft and falls out rapidly,or it grows quickly to attain a fall velocity large enoughto fall out within the updraft; in either case its lifetimeis short. The smaller particles can have longer Lifetimes,. because they can spend a lot of time growing slowly ina region of the storm characterized by both weak updrafts and weak horizontal winds. In this storm, thereis a rather large region with these characteristics in thevicinity of 4 km MSL, near the altitude where the stormand the environmental wind velocities are comparable.Figure 2 shows such a region at the west end of thestorm at 4.8 km MSL; at 3.8 km, a similar region ispresent to the north and east of the main updraft core. Over a local region in a storm in which residenceTABLE 1. Residence times within the storm, 1655 scan, as a function of Do: Case 2, (8.9, 3.6) m s"Residence time (minutes)' Do 0.25 0.5 1.0 1.5 2.0Without fluctuations ITotal accepted trajectories 19 35 17 INumber longer than 40min . 19 11' ' 0 0 0Average residence time(min)* 21.7 15.6 15.3 13.7With fluctuationsTotal accepted trajectories( 10 runs) 935 349 182 51Number longer than 40min 73 000AverageZ time (min) 19.4 16.2 15.4 16.4' For case 1, the number here is zero; for case 3, 27. Lower fallIn this average, the trajectories that stay longer than 40 minutesvelocities, other things being equal, increase the residence time.are counted as 40.times for millimetric particles are long, one might expect substantial accumulation of such particles. Thisseems an interesting possibility with potential importance. It is planned to examine this further, in this andin other storms.It is also interesting that the effect of fluctuations onthe residence times is to make them more nearly independent of size, reducing the average growth timesof the smaller, final-sized particles and lengtheningthose of the larger.As is indicated in the footnote in Table 1, the microphysical specifications are very important to particlelifetimes; as expected, the lower fall velocities producelonger lifetimes.6. Concluding discussion and miscellaneous remarksa. ClassificationOf the storms for which trajectory studies have beenreported, the 12 June 198 1 Montana storm discussedherein is most similar in size and character to thenortheast Colorado storms of 25 July 1976 (Miller etal., 1983) and of 22 July 1976 (Heymsfield et al., 1980; ,Heymsfield, 1983; Foote, 1984). The similarity lies inthe storm size and vigor, but the present storm existedin a more highly sheared environment and had aweaker multicell character with no upwind embryosource. The conclusions of a wide variety of trajectoriesand the ensemble not being very sensitive to microphysical specifications concerning the drag law, werereached both here and for the 22 July storm. In Foote's( 1984) classification, however, while the northeast Colorado storm was OS (open-cell, simple trajectory) thisstorm is perhaps a case of CV (closed cell, various trajectories) since both simple and looping trajectoriesabound and one can easily name other kinds as well.Perhaps the most striking feature of this storm is thegreat variety of embryo source locations and impliedformation mechanisms in this storm, compared to thesomewhat simpler picture gained of each of the northeast Colorado storms.An eventual goal of precipitation trajectory studiesis to classify storms in some way that is meaningful interms of understanding and predicting precipitationefficiency, amount and/or type. Foote (1 984) providesa discussion of this, and gives a suggested classificationscheme in terms of open or closed cells and looping orsimple trajectories. This was expanding on a dichotomyclearly enunciated in two models described in Browning et al. (1976) and Browning and Foote (1976)?The following general remarks are inspired by thisstorm's relatively steady character. While cellularity isthe main basis for the universally applied storm classification, undoubtedly because it is a feature that isoften recognizable with a simple radar, it is questions Neither model is completely original in the paper cited: Foote(1984) gives a good discussion of the extensive literature, whichneedn't be repeated here.1072 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43. No. 11able at this stage of knowledge whether one shouldattempt to use it in a classification intended to be significant for precipitation formation. The difference, bydefinition, between a long-lived multicell storm and along-lived single cell storm is the steadiness of the airflow, usually as signaled by the steadiness of the radarecho pattern. There appears to be a pretty good correlation between the degree of steadiness and the formof the airflow itself (Chisholm and Renick, 1972; Marwitz, 1972a,b,c; an excellent summary in Browning,1977). Also, the differences are beginning to be understood through model studies (Weisman and Klemp,1982). However, few multiple-Doppler flow descriptions have been completed, and there are problematicalcases, as emphasized by Knight (1984). Indeed, thestorm described here is quite steady, yet it lacks mostof the "archetypical" features of a supercell (Browning,1977).The point we wish to make is that, unless the steadiness per se can be shown to be important to precipitation formation, it seems unfortunate (or at the leastpremature) to single out steadiness in a classification.If it is the airflow organization (not the steadiness, butperhaps fairly well correlated with it) that is important,then the distinction should be tied to that organization,not to the steadiness. In this light, two examples arerelevant. Foote (1 984) remarks that in terms of organization the major trajectory type in the multicellWestplains storm is essentially identical to that postulated in the supercell Fleming storm. Knight (1984)shows strong circumstantial evidence that an exceptionally severe hailfall from a very steady, vaulted stormis probably due more to the unsteady flow associatedwith the formation of the vault than to the steadierflow associated with its maintenance.Thus we suggest that attempts at classification of theprecipitation mechanisms of thunderstorms in termsof the cellularity of the storms may be quite misleadingat this stage. More extensive use needs to be made oftime-resolved Doppler wind fields to determine the degree to which the steadiness itself is important. No studyyet has attempted to do this. If steadiness turns out tousually be a minor factor, then a useful classificationcan be sought in terms of relating airflow organizationto trajectory types.b. Directions of further workIt is clear that by far the most important uncertaintyin this whole study has been the specification of theliquid water content. Much of the strong updraft regionin the 12 June 198 1 storm has intense radar echo, evenas low as cloud base (Fig. 2), and the actual depletionof cloud water by precipitation could bring about amajor, systematic modification of the field of liquidwater content, and therefore the trajectory results. Asa first estimate of a more realistic cloud water contentfield, it is planned to estimate the depletion along airparcel trajectories that end at each grid point withinthe storm. The history of the radar echo along an airparcel trajectory as a function of time can be used toobtain such an estimate; and the effect of dilution bymixing might be superimposed using updraft velocity.Even if this procedure fails to make the results lookbetter-they already look better than either author expected-it will be useful as a further sensitivity test onthe effect of realistic, systematic changes in the field ofcloud water content.If this change does improve the realism of the results,it may be possible to investigate the evolution of thisstorm in more detail. The storm changes slowly andsteadily over the 80 minutes of Doppler data, fromhaving an excellent correlation between radar echo intensity and updraft velocity at first, to increasingly moreseparation of the two maxima as time progresses. Toward the end of the study period (1 800), much of theupdraft core is not resolved because of lack of radarecho.A further, possible refinement of the liquid waterfield might be accomplished by modifying the liquidwater content according to the buoyancy field ratherthan the updraft field. The technique is embodied inWeisman and Klemp (1984), in which they show thatin storms in strongly sheared environments, dynamically induced upward accelerations can be comparableto the buoyant acceleration. In such circumstances (andthe present case may be one), a substantial improvement in the liquid water field might result from usingbuoyancy rather than updraft, since not all the airwithin the updraft need have had the same initial thermodynamic characteristics and not a little of it maybe mixed with dry, environmental air.Reliable measurements of liquid water from aircraftwould be a great help for this kind of study, but suchmeasurements in the thunderstorm environment remain difficult and somewhat uncertain. However, thearmored T-28, operated by the South Dakota Schoolof Mines.and Technology, has obtained data in somestorms that have been useful (e.g., Heymsfield, 1983).An ultimate goal of precipitation trajectory studiesis to enable generalizations to be drawn about precipitation formation in different types of storms. A systematic method that is adequately realistic may notexist yet, except perhaps for dealing with the largesthail. If the problems with estimating the field of liquidwater content and accounting for particle concentrations can be resolved, then the systematic applicationof the backward-trajectory scheme used here might bean aid in comparing different storms. It does lend itselfto relatively objective judgements of the relative importance of different trajectory types.c. Final remarksA lot of the CCOPE data on the 12 June 198 1 stormhave neither been used nor mentioned. The dualwavelength radar system (10 and 3 cm) was in use, andin fact the 3-cm data were useful in identifying seriousside-lobe contamination in the 10-cm data. Hail signal(see Rinehart and Tuttle, 1982) stronger than 3 dB andwith temporal and spatial continuity was present Onlyin two separate, five-minute periods at approximatelyI JUNE 1986CHARLES A. KNIGHT AND KEVIN R. KNUPP107317 10 and 1720, after the main time of study here. Tothe extent that lack of a hail signal indicates absenceof hail, the accepted trajectories leading to 1.5 and 2.0cm particles must be treated skeptically. This wouldsupport our suspicion that the liquid water content maybe substantially overestimated in the present study.Cloud-to-ground lightning strikes were monitoredand located by a system operated by SUNY-Albany(R. Orville), which had a real-time readout in theCCOPE operations room. The system worked well on12 June, as indicated by response to a storm earlierand farther north than the storm discussed here, andby recording many strikes from a very large and severestorm later, to the southwest. To everyone's astonishment, the system indicated only one or two cloud-toground strikes from the storm studied here. It wouldbe interesting to try to correlate lightning occurrencewith various radar parameters, throughout the CCOPEdata set.Acknowledgments. We thank the many CCOPEparticipants for the data set. Ms. Joanne Parrish programmed and ran the trajectory calculations. Ms.Frances Huth typed several drafts of this paper. Whileseveral people made valuable comments and suggestions at different stages, we thank especially Dr. AndrewWeinheimer, Dr. G. Brant Foote, Dr. Warren D. King,Dr. Stephan Nelson, and two anonymous reviewersfor helpful criticisms. This work was supported in partby the National Science Foundation under GrantATM-85-12480.REFERENCESBrandes, E. A., 198 1 : Fine structure of the Del City-Edmond tornadicBrowning, K. A., 1963: The growth of large hail within a steadymesocirculation. Mon. Wea. Rev., 109,635-647.updraft. Quart. J. Roy. Meteor. SOC., 89, 490-506.Review ofHail Science and Hail Suppression. Meteor. Monogr.,No. 38, G. B. Foote and C. A. Knight, Eds., Amer. Meteor.SOC., 1-43.storms and some implication for hail suppression. Quart. J. Roy.Meteor. SOC., 102, 499-533.F. H. Merrem, D. J. Musil, E. L. May and W. R. Sand, 1976:Structure of an evolving hailstorm: Part V. 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The growth trajectories of precipitation particles that attain diameters from 0.5 to 2.0 cm are modeled within the wind field of a small, relatively steady-state, southeastern Montana thunderstorm. The trajectories are calculated backwards, from systematic arrays of particles of specified sizes at a level near cloud base. Using a simple set of criteria for rejecting the obviously impossible trajectories, the patterns of accepted trajectory end-points are compared with the radar echo patterns. Good agreement lends credence to the qualitative aspects of the trajectories. For a given size of precipitation particle, the method helps one to assign different trajectory types to specific regions within the horizontal plant on which the calculations were started. The relative importance of the different types of trajectories can thus be estimated. Particle origin mechanisms are discussed in terms of the regions in which the trajectories are found to start. The variety of successful trajectories leading to 1 cm and larger hail in a storm of considerable structural simplicity is noteworthy.
Sensitivity tests indicate that the liquid water content is by far the most important specification in this framework. Ongoing work is directed toward improving this specification and deriving estimates of particle concentration.