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VOL. 60, NO. 6 15 MARCH 2003J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

q 2003 American Meteorological Society 781

A Scale-Discriminating Vorticity Budget for a Mesoscale Vortex in a Midlatitude,Continental Mesoscale Convective System

JASON C. KNIEVEL

National Center for Atmospheric Research, Boulder, and Department of Atmospheric Science,Colorado State University, Fort Collins, Colorado

RICHARD H. JOHNSON

Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

(Manuscript received 18 January 2002, in final form 23 September 2002)

ABSTRACT

The authors employ data from the NOAA Wind Profiler Network to present a scale-discriminating vorticitybudget for a mesoscale convective vortex (MCV) that was generated by a mesoscale convective system (MCS)in Oklahoma and Kansas on 1 August 1996.

A spatial bandpass filter was used to divide observed wind into mesoscale and synoptic components. Thenthe authors sought sources and sinks of vorticity in those two components over 9 h of the MCV’s lifetime.

The vorticity budget reveals that both the mesoscale and synoptic winds supplied significant vorticity to theMCV. The vortex’s origin could not be proved, but data weakly suggest that tilting may have been mostlyresponsible. Convergence of absolute vorticity by the mesoscale wind was the reason the MCV grew deeperand stronger as the MCS weakened. Finally, tilting of synoptic and mesoscale vorticity by gradients in mesoscalevertical motion was responsible for a secondary deepening of the MCV as the MCS dissipated.

The budget suggests that, if the MCV of 1 August 1996 is representative, completely realistic simulations ofMCVs should include planetary vorticity and a plausible, three-dimensionally heterogeneous synoptic wind.

1. Introduction

In this paper we present a vorticity budget for a me-soscale convective vortex (MCV) generated by a me-soscale convective system (MCS) that traversed Kansasand Oklahoma on 1 August 1996. The budget discrim-inates between sources and sinks of vorticity within themesoscale wind of the MCS and within the synopticbackground wind.

a. Background

MCSs often generate MCVs in the lower and middletroposphere. An MCV’s tangential wind speed is of or-ders 1 and 10 m s21, its diameter of order 100 km, andits lifetime of orders 1 and 10 h. MCVs organize MCSson the mesoscale (Menard and Fritsch 1989; Brandes1990) and can serve as the primary dynamical linkamong serial MCSs (Raymond and Jiang 1990; Fritschet al. 1994; Trier et al. 2000).

If friction is ignored, vertical vorticity within an MCVmust originate from some combination of (a) horizontal

Corresponding author address: Dr. Jason Knievel, NCAR, P.O.Box 3000, Boulder, CO 80307-3000.E-mail: [email protected]

advection of absolute vorticity, (b) vertical advection ofrelative vorticity, (c) convergence of absolute vorticity,(d) tilting of horizontal vorticity by horizontally varyingvertical wind, and (e) horizontal baroclinity. This is ap-parent upon examination of the equation for the localchange in relative vertical vorticity of an inviscid fluidon an f plane:

]z ]z5 2[v · =(z 1 f )] 2 w 2 [(z 1 f )= · v]1 2]t ]z

| | | | | || | |a b c

]w ]w1 j 1 h 1 [J (p, a)] , (1)xy1 2]x ]y

| | | || |d e

wherein z(j, h, z) is relative vorticity, v(u, y) is hori-zontal wind, w is vertical wind, f is the Coriolis pa-rameter, and Jxy(p, a) is the two-dimensional Jacobianof pressure, p, and specific volume, a. (Letters assignedto the terms correspond to the enumeration earlier inthe paragraph.) Baroclinity is weak near MCVs and canbe ignored without losing much accuracy (Skamarocket al. 1994; Cram et al. 2002). (Henceforth, vorticity

782 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

means relative vertical vorticity, and divergence, con-vergence, and shear mean horizontal divergence, hori-zontal convergence, and vertical shear, unless otherwisestated.) In a Lagrangian sense, advections only redis-tribute vorticity, they do not create it, so analyses ofvorticity in MCVs tend to emphasize divergence andtilting, terms (c) and (d) in (1). Contributions from bothsources frequently are large (e.g., Brandes 1990; Chongand Bousquet 1999).

A positive contribution to vorticity can come fromconvergence in the middle troposphere within the strat-iform region of an MCS, where existing planetary andsynoptic vorticity can be concentrated (Bartels and Mad-dox 1991; Johnson and Bartels 1992; Skamarock et al.1994). Some of this convergence is due to the atmo-sphere’s balanced response to heating and cooling byphase changes in water within the stratiform region ofan MCS (Hertenstein and Schubert 1991). The strengthof this balanced response depends on the Rossby radiusof deformation:

NHl 5 , (2)R 1/2 21 1/2(z 1 f ) (2VR 1 f )

wherein N is the Brunt–Väisällä frequency, H is thescale height of the disturbance, z is the vertical com-ponent of relative vorticity, f is the Coriolis parameter,and V is the wind’s rotational component, of which Ris radius of curvature (Schubert et al. 1980; Frank 1983).It is generally appropriate to choose values for N, z, f ,and V that are averaged over the region diabaticallyheated by moist convection. In an environment of 100%relative humidity, rising air is heated by condensationas well as cooled by expansion, so N should be replacedby Nm, as explained by Durran and Klemp (1982). Whensources of diabatic heating are larger than lR, moreenergy is retained in balanced, vortical flow near anMCS than is transmitted to the far field by gravity wavesand buoyancy rolls; the opposite occurs when sourcesof diabatic heating are smaller than the Rossby radius(Schubert et al. 1980; Mapes 1993). An MCS can shrinkits local Rossby radius by saturating the stratiform re-gion, which means that a potentially lower-valued Nmreplaces N, and by generating a vortical circulation,which increases 2VR21 (Schubert and Hack 1982; Cot-ton et al. 1989; Chen and Frank 1993).

Tilting’s positive contribution can come from hori-zontally varying vertical motion in an MCS’s mesoscaleupdraft or downdraft that tilts vorticity equated withvertically and horizontally sheared wind (Davis andWeisman 1994). Shear exists in both the wind of anMCS and the wind of the system’s environment, butinitial studies of tilted vorticity in MCVs emphasizedonly environmental shear (e.g., Biggerstaff and Houze1991) or did not formally differentiate between the twoshears (e.g., Brandes 1990; Zhang 1992). More recentstudies demonstrated that vertical shear in MCSs is alsoan important source of vorticity within MCVs (e.g.,Chong and Bousquet 1999; Bousquet and Chong 2000).

b. Objective and motivation

Most empirical vorticity budgets for MCVs fall intotwo categories. In the first are studies of vorticity onthe scale of an MCS and MCV; these are usually basedon Doppler radars and/or research sounding networks(e.g., Biggerstaff and Houze 1991; Johnson and Bartels1992; Chong and Bousquet 1999). In the second arelarger-scale budgets based on the nation’s operationalsounding network (e.g., Bartels and Maddox 1991).Studies in both categories are usually valid only for afew hours or for a single stage of an MCV because ofthe difficulty of obtaining detailed kinematical data overa substantial fraction of an MCV’s lifetime.

Before now, no empirical vorticity budget for anMCV simultaneously investigated circulations within anMCS as well as those within the MCS’s larger-scaleenvironment. The MCV of 1 August 1996 presented anopportunity for just such an investigation when the vor-tex spent the majority of its lifetime within the densestpart of the National Oceanic and Atmospheric Admin-istration (NOAA) Wind Profiler Network. In a closelyrelated paper (Knievel and Johnson 2002), we examinedthe life cycle and kinematics of the MCS that generatedthe MCV. Our objective in this paper is to extend thatresearch by presenting for an MCV the first vorticitybudget that explicitly comprises terms for sources andsinks of vorticity within an MCS, its environment, andcombinations of the two. The budget has the added ad-vantage of encompassing 9 h of the MCV’s lifetime.

2. Data and methods

Observations are from the National Weather Service(NWS), remote sensors operated by NOAA, the 1996Enhanced Seasonal Observing Period (ESOP-96) of theGlobal Energy and Water Cycle Experiment’s (GEW-EX’s) Continental-Scale International Project (GCIP),and the Oklahoma Climatological Survey’s OklahomaMesonet (Brock et al. 1995).

The previously published kinematical analysis of theMCS of 1 August 1996 (Knievel and Johnson 2002)provides details about the data and some of our methods,so herein we emphasize only the methods integral tounderstanding the vorticity budget and related calcu-lations.

a. Objective analyses and bandpass filtering

To produce gridded fields of total wind, u (u, y, w),we used a two-pass Barnes analysis (Barnes 1973; Kochet al. 1983) on data from the NOAA Wind Profiler Net-work (NPN; Fig. 1). Grid points were 75 km apart, thecutoff radius was 750 km, and the response functionwas chosen to capture 90% of the signal of phenomenawith wavelengths of 300 km, which is twice the averagedistance between profilers in the densest part of the NPN(total curve in Fig. 2). Less than 10% of the signal of

15 MARCH 2003 783K N I E V E L A N D J O H N S O N

FIG. 1. Sites of observations above the ground. NOAA wind pro-filers are marked by black squares. The inset box shows the locationsof GCIP sounding sites.

FIG. 2. Response functions of the Barnes analyses.phenomena with wavelengths shorter than 85 km wascaptured, so virtually no coherent convective signal ex-ists in the analyzed data.

To isolate partially the mesoscale kinematics of theMCS from the synoptic kinematics, we employed a sec-ond Barnes analysis that, together with the first, actedas a bandpass filter (Maddox 1980). The synoptic back-ground wind, ũ(ũ, , w̃), was approximated with dataỹfiltered to include 90% and 0.09% of the signals ofphenomena with wavelengths of 1600 and 300 km, re-spectively (synoptic curve in Fig. 2). This is the samefiltration that Maddox (1980) used for synoptic features.The mesoscale perturbation in wind, û(û, , ŵ), wasŷapproximated by subtracting the synoptic backgroundwind from the total wind (mesoscale curve in Fig. 2).In summary, u(u, y, w) 5 ũ(ũ, , w̃) 1 û(û, , ŵ).ỹ ŷ

The bandpass filter did not completely isolate the me-soscale and synoptic winds from each other, especiallyat scales near where the two response curves cross (Fig.2). However, the wavelengths of the MCS and MCV of1 August 1996 were near the peak of the mesoscaleresponse function, ;400 km, where the synoptic partof the filter was quite insensitive.

b. Divergence, vorticity, and vertical velocity

Calculations of divergence and vorticity are from cen-tered finite differences of objectively analyzed fields ofu and y components of the wind. We found this methodsufficiently accurate when tested against line integralsof tangential and normal components of wind calculatedaround the perimeter of triangles whose vertices werethe profilers (Ceselski and Sapp 1975). Vertical velocityis from the kinematical method with a linear correctionto density-weighted divergence (O’Brien 1970), where-

in w 5 0 at 1000 m above the tropopause and at 500m above ground level (AGL).

c. Averages over the stratiform region

Although the NPN provided a much more resolveddataset than would have been available from operation-ally launched radiosondes alone, missing data still madeit impossible to perform detailed hourly analyses at ev-ery grid point because large spatial gaps greatly reducedthe accuracy of the Barnes analyses at a few individualtimes. We mitigated this problem by examining objec-tively analyzed data that were then temporally and spa-tially averaged over 3 h and a 28 3 28 area centered onthe MCV, whose radius of maximum wind was approx-imately 0.758–1.508 latitude (83–167 km).

Official observations from the NPN for a time t areof wind recorded over the hour ending at t. So, forexample, an average of NPN observations from 1000,1100, and 1200 UTC, such as we calculated, is an av-erage of wind recorded from 0901 to 1200 UTC.

d. Vorticity budget

As mentioned in section 1, for a vorticity budget ofwind above the boundary layer and any cold pools, itis reasonable to dismiss baroclinity, represented by theJacobian in (1). The remaining four terms in (1) wereretained for the budget.

For our case, the local derivative on the left side of(1) is for a 28 3 28 area repositioned every hour so itwas centered on the midtropospheric part of the MCV.Mesoscale wind in the middle troposphere was virtually


the same as a system-relative wind (i.e., wind in a frameof reference that moved with the MCV) because thetranslational motion of the MCV was due primarily toadvection by the background synoptic wind in the mid-dle troposphere, which is common (e.g., Zhang andFritsch 1988; Johnson and Bartels 1992; Trier et al.2000). Over the detectable lifetime of the MCV, zonaland meridional components of the synoptic backgroundwind at the three-dimensional center of the MCV were7.7 and 28.1 m s21, respectively. The correspondingcomponents of the MCV’s average translational velocitywere 7.4 and 28.1 m s21. Therefore, once the synopticwind was removed from the total wind, the frame ofreference moved in the middle troposphere with theMCV, and the circulation of the mesoscale perturbationin wind within the MCV was approximately closed(Knievel and Johnson 2002).

We calculated on a grid the vorticity budget for themesoscale perturbation in wind, for the synoptic back-ground wind, and for the sum of the two, the total wind.This sum is not truly a total wind in that it does notcontain fully sampled contributions from subgrid phe-nomena such as eddies from cumulonimbi. First, theNPN is insufficiently dense to record unaliased obser-vations of such phenomena. Second, filtering by theBarnes routine excluded nearly all sources and sinks ofvorticity with wavelengths smaller than 85 km. Yet phe-nomena this small undoubtedly contributed to temporalchanges in vorticity (Weisman and Davis 1998), andthose contributions probably translated to larger scalesto which the Barnes routine was sensitive (Esbensen1993). Fortunately, eddy fluxes of momentum in MCSsare generally less in stratiform regions than in convec-tive lines (Gallus and Johnson 1992), and our vorticitybudget is limited to the former.

Because of such unresolved sources and sinks of vor-ticity, the vorticity budget has a residual, which alsoincludes observational errors. When written in terms ofthe resolved synoptic component [designated by )],(˜the resolved mesoscale component [designated by ()],and the residual Z, (1) becomes

]z ](z̃ 1 ẑ )5 2(ṽ 1 v̂) · =(z̃ 1 f 1 ẑ ) 2 (w̃ 1 ŵ)

]t ]z

](w̃ 1 ŵ)2 (z̃ 1 f 1 ẑ )= · (ṽ 1 v̂) 1 (j̃ 1 ĵ )

]x

](w̃ 1 ŵ)1 (h̃ 1 ĥ) 1 J ( p̃ 1 p̂, ã 1 â) 1 Z.xy]y

(3)

We vertically smoothed terms in the budget with a five-point, center-weighted running mean.

Often the resolved terms on the right side of a budgetthat is in the form of (3) together nearly cancel theresidual, leaving the local tendency as a small differencein this near cancellation (e.g., Reed and Johnson 1974).Therefore, the magnitude and even the sign of the local

tendency, ]z/]t, is very sensitive to errors in the hori-zontal derivatives that compose the resolved terms. Thissensitivity is especially troublesome if one uses the localtendency to predict vorticity. We made no such predic-tions because observations of vorticity were availableevery hour for our analysis. Even so, it is valuable toassess how the resolvable part of the local tendencydiffers when calculated with a budget based on a math-ematically simpler form of the vorticity equation thanthe form on which we based (3).

One such equation, used in vorticity budgets by Davisand Weisman (1994) and Weisman and Davis (1998),among others, may be written most succinctly as

]z5 = · K, (4)

]t

in which friction and baroclinity are ignored, and

]vK 5 w k 3 2 v(z 1 f ), (5)1 2]z

wherein k is the vertical unit vector and all the othersymbols have meanings as in (3). When (4) and (5) arecombined, the explicit terms are

]z ] ]y5 2 w 1 u(z 1 f )[ ]]t ]x ]z

] ]u2 w 1 y(z 1 f ) . (6)[ ]]y ]z

We checked the results of (3), which we call the con-ventional form of the vorticity budget, with (4), whichis sometimes called the flux form, but which we callthe divergence form of the vorticity budget. Results ofthe check appear in section 5.

e. Rossby radius of deformation

Calculations of the Rossby radius of deformation, lR,were hampered by scant thermodynamical data. One ofonly two GCIP radiosondes from which data were notlost during penetration of the stratiform region waslaunched at 1800 UTC from Morris, Oklahoma (B5).From that sounding (not shown) we calculated N in (2)over a depth of 3 km to be 1.05 3 1022 s21, which weused in calculations for the stratiform region at othertimes. The resultant value of 31.5 m s21 for NH is veryclose to the fixed 30.0 m s21 used by Cotton et al. (1989)in their evaluation of changes in lR over the lifetime ofa composite mesoscale convective complex (MCC), aspecific type of large MCS. Although our calculation oflR was for a part of the stratiform region, the lowertroposphere was subsaturated there, so we used N in-stead of the Nm mentioned in section 1a.


3. Overview of vorticity in the MCS and MCV

The MCS, which comprised a leading convective lineand a trailing stratiform region, formed at 0430 UTCon 1 August 1996 and had completely dissipated by0315 UTC on 2 August (Knievel and Johnson 2002).The first sign of an MCV was at 0715 UTC on 1 August,and by 0315 UTC on 2 August the MCV was no longerdetectable in radar, satellite, or profiler data. Figure 3depicts the MCS as it appeared in composite radar sum-maries. Even in these static schemata, little imaginationis needed to envision the vortical flow that defined theMCV and shaped the MCS’s stratiform region.

Over the 9 h we examined, 0900 to 1800 UTC on 1August (which we call the period of detailed analysis),the MCV deepened and strengthened as the MCS ma-tured and dissipated. Eventually the vortex occupiedalmost the entire troposphere, perhaps even reaching theground (Knievel and Johnson 2002).

Positive vorticity in the MCV seemed to originate atabout 3 km above mean sea level (AMSL), in a non-divergent layer of weak ascent (Fig. 4), and from 0900to 1200 UTC this was approximately the altitude of itsaverage maximum vorticity (Fig. 5a). Then the top ofthe vortex and the level of maximum vorticity both as-cended between 1200 and 1500 UTC (Fig. 5b) as thevortex strengthened and the stratiform region dominatedthe MCS, judging from the profile of vertical motion(Houze et al. 1989). After these ascensions, the altitudeof maximum vorticity remained approximately fixed at6 km AMSL for the remainder of the period of detailedanalysis, during which time the vortex’s top reached 13km AMSL (Fig. 5c).

The long-lived simulated MCV of Zhang and Fritsch(1988) behaved similarly: the height of maximum vor-ticity varied little for the first 2 h of the MCV’s life,rose quickly as the MCV strengthened, then varied littleafter that. Conversely, Chen and Frank (1993) simulatedan MCV whose vorticity maximum descended, not as-cended, with time. Few empirical studies recount tem-poral variations of vorticity within an MCV, but amongthese few Menard and Fritsch (1989) did find that max-imum vorticity ascended with time in the MCV of 6–7July 1982, which was the MCV simulated by Zhang andFritsch (1988).

The MCV of 1 August 1996 was deeper than manyMCVs, but not all. The simulated MCVs of Chen andFrank (1993) and Rogers and Fritsch (2001), respec-tively, had tops near and above 200 hPa. (In our case,200 hPa was approximately 12.4 km AMSL.) The ob-served MCVs of Brandes (1990) and Bousquet andChong (2000) had tops near 11 km AMSL.

The shallow, weak, negative vorticity above the ma-ture MCV of 1 August 1996 (Fig. 5c) is the signatureof planetary vorticity’s effect on divergent outflow fromhigh pressure in the upper troposphere (e.g., Brandes1990; Johnson and Bartels 1992; Bousquet and Chong2000).

A more thorough discussion of divergence and ver-tical motion within the MCV appeared in our earlierpaper (Knievel and Johnson 2002). Please see it formore detailed commentary on Fig. 5.

4. Vorticity budget

According to our application of (3), convergence, tilt-ing, and unresolved effects contributed the most to netchanges in the MCV as it matured, and prominent sourc-es and sinks of vorticity existed in both the synopticand mesoscale components of the wind. (The vorticity,divergence, and vertical motion presented in Fig. 5 mayaid in the interpretation of the budget in the followingsection.)

a. Total wind

In the total wind, there were only two positive sourcesof vorticity in the lower troposphere at the altitude ofthe developing MCV between 0900 and 1200 UTC:tilting (Fig. 6b) and unresolved effects (Fig. 7). Becausethe vortex had already formed by the start of the periodof detailed analysis, we could not conclusively deter-mine the source of vorticity in the incipient vortex. Evenso, Fig. 6b weakly suggests that tilting may have playedthe largest role on resolved scales, which would be con-sistent with the study by Zhang (1992), whose simulatedMCV began primarily from tilting, then in maturitystrengthened from convergence.

The MCV of 1 August 1996 grew deeper and strongerbetween 1200 and 1500 UTC, primarily from conver-gence of positive absolute vorticity in the middle tro-posphere (Fig. 6c). Planetary and relative vorticitiescontributed almost equally (not shown). If not for di-vergence of planetary vorticity in the upper troposphere,the tendency due to divergence there at 1500 UTCwould have been positive as well, because upper-tro-pospheric relative vorticity was negative (Fig. 5b). In-deed, because divergence and relative vorticity wereroughly anticorrelated about zero in the middle and up-per troposphere (Fig. 5b), any deep layers of stronglynegative tendency due to divergence at those altitudesmust have been from divergent wind acting on planetaryvorticity, because divergence of negative relative vor-ticity and convergence of positive relative vorticity can-not produce a negative tendency. Davis and Weisman(1994) alluded to this. At the same time that conver-gence of absolute vorticity generated relative vorticityin the lower and middle troposphere, the mesoscale up-draft advected that vorticity upward (Fig. 6d). However,positive vertical advection of vorticity was over-whelmed by all the other resolved sinks. In particular,horizontal advection decreased vorticity from the lowerthrough the upper troposphere, which is partly a resultof the way vorticity was averaged; as long as the 28 328 area of averaging remained centered on the maximumvorticity in the MCV, any horizontal advection into or


FIG. 3. Schemata of composite base-scan radar reflectivity on 1 Aug 1996. Times are (a) 0345, (b) 0545, (c) 0715, (d) 0830, (e) 1100, (f) 1200,(g) 1500, and (h) 1800 UTC. Contours are at 15, 30, and 45 dBZ. Only reflectivity due to the MCS and nearby cumulonimbi is shown.

out of that area would necessarily produce a negativetendency. At 1500 UTC, sources that compose part ofthe residual appear to have acted in concert with verticaladvection above the MCV but at the altitude of the MCVstrongly opposed the positive tendency from conver-gence (Fig. 7).

Convergence of absolute vorticity between 1200 and

1500 UTC was likely aided by the existent MCV. Asexplained in section 1a, when sources of diabatic heatingare larger than the Rossby radius of deformation, lR,more energy is retained in balanced, vortical flow nearan MCS than is transmitted to the far field by gravitywaves and buoyancy rolls (Mapes 1993; Schubert et al.1980). The transition to such balanced, vortical flow


FIG. 4. Relative vorticity (solid, in 1025 s21), divergence (dashed,in 1025 s21), and vertical velocity (dotted, in 1022 m s21) of the totalwind at 1000 UTC 1 Aug 1996. Profiles are for a 28 3 28 area centeredon the MCV. Unlike in Fig. 5, profiles are not for 3-h averages. Databelow 1750 m AMSL are not plotted. The levels of 08C in the en-vironment and of the tropopause are marked along the right side.

FIG. 5. Relative vorticity (solid, in 1025 s21), divergence (dashed,in 1025 s21), and vertical velocity (dotted, in 1022 m s21) of the totalwind on 1 Aug 1996. Profiles are for a 28 3 28 area centered on theMCV, averaged over 3 h ending at the times labeled. Data below1750 m AMSL are not plotted. The levels of 08C in the environmentand of the tropopause are marked along the right side of each panel.

involves convergence. On 1 August 1996 that transitionwas facilitated by the MCV’s vorticity, which reducedlR. At 0800 UTC, close to the time when a vorticalcirculation was first visible in loops of composite re-flectivity, lR was 276 km, which is close to the 280 kmcalculated by Chen and Frank (1993) and the 300 kmcalculated by Cotton et al. (1989) for MCS environ-ments. (Calculations of the Rossby radius are explainedin section 2e.) By 1200 UTC, lR had shrunk to 136 kmdue to the increase in background vorticity. It stayedclose to that value through 1500 UTC, the interval ofmaximum strengthening of the MCV. The radius of max-imum wind we estimated for the MCV over multiplehours is 0.758–1.508 latitude (83–167 km). The size ofthe stratiform region was difficult to measure precisely;the major axis was perhaps 350 km long during theasymmetric stage of the MCS, giving a pseudoradius of175 km. This is slightly larger than lR, so a sizablefraction of the atmosphere’s response to heating wasretained near the MCS as convergent, vortical flow inthe middle troposphere between 1200 and 1500 UTC.Rogers and Fritsch (2001) found a very similar rela-tionship among preexisting vorticity, static stability, lo-cal Rossby radius, and vortex spinup in their simulationof a long-lived MCV.

In the MCS of 1 August 1996, tilting was the primarysource of the positive, upper-tropospheric vorticity thatfurther deepened the MCV during the final 3 h of theperiod of detailed analysis (Fig. 6f). Tilting and con-vergence were the primary sources of vorticity in thelower troposphere. Some of the strength of the MCVin the middle troposphere was maintained by effectsrepresented by the residual and by convergence of ab-solute vorticity, because convergence, although weak-ening, continued from 6.5 to 11.0 km AMSL (Fig. 5c).Three-dimensional advection was generally a sink of

vorticity at 6 km AMSL, the altitude of maximum vor-ticity in the MCV (Figs. 6e,f).

A few general properties of the vorticity budget forthe total wind deserve mention. In agreement with ob-servations by Chong and Bousquet (1999) and others,tilting and vertical advection of vorticity were veryroughly anticorrelated about zero (Figs. 6b,d,f). Only


FIG. 6. Resolvable part of the vorticity budget for the total wind on 1 Aug 1996. (a), (c), (e) Horizontal advection(dashed) and horizontal divergence (dot–dashed). (b), (d), (f ) Vertical advection (dotted) and tilting (dot–dot–dashed). Thesum of all four terms (heavy solid) appears on each panel. Profiles are for a 28 3 28 area centered on the MCV, averagedover 3 h ending at the times labeled.

when this anticorrelation broke down did tilting andvertical advection play large, net roles in the local ten-dency. The often similar anticorrelation between hori-zontal advection and divergence of vorticity (e.g., Bran-des and Ziegler 1993) was weaker, but still evident attimes—in the lower and middle troposphere from 1500to 1800 UTC, for example (Fig. 6e). Unresolved effectsand observational errors represented by the residualwere just as large as those explicitly resolved in thebudget (cf. Figs. 6 and 7), which strongly suggests thatregions of persistent convective-scale circulations sig-

nificantly altered atmospheric vorticity on the meso-scale.

Much of the empirical research into this upscale com-munication of vorticity in masses of moist convectionhas focused on the large-scale effects of clusters of cu-muli in the Tropics (e.g., Esbensen et al. 1982; Tollerudand Esbensen 1983; Sui and Yanai 1986). Figure 7 gen-erally does not resemble profiles of residuals calculatedfor such tropical cloud clusters, but attempts to explainthe lack of resemblance are beyond the scope of thispaper.


FIG. 7. Vorticity tendency due to the residual in the total wind at1200 UTC (short dashed), 1500 UTC (long dashed), and 1800 UTC(solid) on 1 Aug 1996. Profiles are for a 28 3 28 area centered onthe MCV, averaged over 3 h ending at the times labeled.

FIG. 8. Vorticity tendency due to components of horizontal advection on 1 Aug 1996. Terms are as follows: advectionof mesoscale perturbation in vorticity by the mesoscale perturbation in wind (solid) and by the synoptic wind (dashed),and advection of synoptic vorticity by the mesoscale perturbation in wind (dashed–dotted) and by the synoptic wind(dotted). Profiles are for a 28 3 28 area centered on the MCV, averaged over 3 h ending at the times labeled.

Upscale communication of vorticity has also been thesubject of numerical studies. For example, in simula-tions by Montgomery and Enagonio (1998), clusters ofcumulonimbi embedded within a vortex were able tostrengthen a midtropospheric vortex through inwardfluxes of angular momentum. The authors used potentialvorticity to represent forcing by moist convection. Ineffect, the existing vortex was cyclonically acceleratedbecause it assimilated positive potential vorticity andexpelled negative potential vorticity. Because the mag-nitudes and distributions of potential vorticity realisti-cally approximated both an MCV (radius of maximumwind was 200 km and midtropospheric tangential ve-locity was 5 m s21) and the moist convection near sucha vortex, it is reasonable to assume that the results ofMontgomery and Enagonio are relevant to MCVs suchas that of 1 August 1996.

b. Discrimination between synoptic and mesoscalewinds

When terms in the vorticity budget, (3), are itemizedaccording to the synoptic background wind (terms withtildes) and the mesoscale perturbation to that back-ground wind (terms with carets), the budget exemplifiescommonly observed—in some cases defining—traits ofsynoptic and mesoscale motions. First, synoptic and me-soscale horizontal velocities were similarly large (Kni-evel and Johnson 2002), so contributions to vorticityfrom ṽ(ũ, ) and v̂(û, ) within the budget’s terms wereỹ ŷsimilarly large. Second, synoptic vertical velocity, w̃,was much smaller than mesoscale vertical velocity, ŵ,because synoptic divergence, = · ṽ, was much smallerthan mesoscale divergence, = · v̂. Accordingly, terms in(3) with w̃ and = · ṽ contributed very little to the budget.Third, as long as planetary vorticity, f , is consideredpart of synoptic vorticity, synoptic and mesoscale vor-ticities, 1 f and , were of the same magnitude, andz̃ ẑin some terms contributed large values to the budget.Gradients in synoptic vorticity were much smaller thangradients in mesoscale vorticity, though. Finally, shearsin synoptic and mesoscale winds were similarly large,so tilting of both ( , ) and ( , ) contributed sim-h̃ j̃ h̃ ĥ ĵ ĥilarly large values to vorticity tendency. The term-by-term analysis of the vorticity budget in the followingsections makes these generalities more meaningful.

1) TENDENCY FROM HORIZONTAL ADVECTION

Nearly all the vorticity tendency from horizontal ad-vection was due to advection of mesoscale vorticity bythe synoptic wind (Fig. 8). The reason horizontal ad-vection of mesoscale vorticity by the mesoscale windwas so small is that, in the predominantly vortical flowof the MCV, the largest component of the mesoscalewind tended to be orthogonal to the gradient of meso-


FIG. 9. Vorticity tendency due to components of vertical advection on 1 Aug 1996. Terms are as follows: advectionof mesoscale perturbation in vorticity by the mesoscale perturbation in wind (solid) and by the synoptic wind (dashed),and advection of synoptic vorticity by the mesoscale perturbation in wind (dashed–dotted) and by the synoptic wind(dotted). Profiles are for a 28 3 28 area centered on the MCV, averaged over 3 h ending at the times labeled.

FIG. 10. Vorticity tendency due to components of horizontal divergence on 1 Aug 1996. Terms are as follows: divergenceof mesoscale perturbation in vorticity by the mesoscale perturbation in wind (solid) and by the synoptic wind (dashed),and divergence of synoptic vorticity by the mesoscale perturbation in wind (dashed–dotted) and by the synoptic wind(dotted). Profiles are for a 28 3 28 area centered on the MCV, averaged over 3 h ending at the times labeled.

scale vorticity [evident in Fig. 4 of Knievel and Johnson(2002)], so the dot product in the horizontal advectionterm was small even though the vectors in the term werelarge. Gradients of synoptic vorticity were too small topermit much horizontal advection, even though synopticvorticity was as large as mesoscale vorticity because weincluded planetary vorticity in the former.

Because larger vertical shears generally lead to short-er-lived MCVs, differential advection seems to be onemechanism that destroys the vortices (Trier et al. 2000).Figure 8 suggests that such differential advection ismostly by environmental wind, not by wind within themesoscale circulations of MCSs. In the case of the MCVwe studied, this differential advection arose not simplybecause of the horizontal translation of vertically vary-ing mesoscale vorticity, but also because of verticallyvarying horizontal synoptic wind. [See Fig. 9 of Knieveland Johnson (2002) for depictions of the synoptic wind.]

2) TENDENCY FROM VERTICAL ADVECTION

Nearly all the vorticity tendency from vertical ad-vection was due to advection of mesoscale vorticity bythe mesoscale wind (Fig. 9). Not surprisingly, the onlyvertical motions strong enough to contribute much tovertical advection were in the mesoscale field. Synopticvorticity, while large, did not lead to large vertical ad-vection even by the mesoscale wind because verticalgradients of synoptic vorticity were small.

3) TENDENCY FROM DIVERGENCE

No component in the vorticity tendency from hori-zontal divergence was negligibly small, although twoof the four components were dominant (Fig. 10). From1200 to 1500 UTC, when the MCV underwent the great-est deepening and strengthening, vorticity in the MCV


FIG. 11. Vorticity tendency due to components of tilting on 1 Aug 1996. Terms are as follows: tilting of vertical shearof the mesoscale perturbation in wind by the horizontal gradient of mesoscale perturbation in vertical motion (solid)and by the horizontal gradient of synoptic vertical motion (dashed), and tilting of vertical shear of the synoptic windby the horizontal gradient of mesoscale perturbation in vertical motion (dashed–dotted) and by the horizontal gradientof synoptic vertical motion (dotted). Profiles are for a 28 3 28 are centered on the MCV, averaged over 3 h ending atthe times labeled.

was generated mostly, and nearly equally, by conver-gence of mesoscale vorticity and convergence of syn-optic vorticity, both by the mesoscale wind. Tendencydue to convergence of synoptic vorticity by the synopticwind was at least a factor of 1/2 smaller than the dom-inant terms, and tendency due to convergence of me-soscale vorticity by the synoptic wind was slightlysmaller yet, especially in the upper troposphere.

It is through convergence that vorticity from short-wave troughs near an MCS would play a role in gen-erating an MCV. The size of a typical shortwave troughwould put its vorticity into both the synoptic and me-soscale regimes, as we have approximated them. Ac-cording to the general conclusions one can draw fromFig. 10, the primary concentrator of vorticity in short-wave troughs is probably the mesoscale wind.

4) TENDENCY FROM TILTING

Finally, the only two components that contributed ap-preciably to the vorticity tendency from tilting weretilting of both synoptic and mesoscale vorticity by hor-izontally varying mesoscale up- and downdrafts (Fig.11). Synoptic drafts were too weak, and their horizontalvariations too small, to significantly tilt tubes of hori-zontal vorticity. Tilted horizontal vorticity consistentlycontained large synoptic as well as mesoscale compo-nents. This differs somewhat from the study by Davisand Weisman (1994), in which they found that tiltingof environmental shear was dominant early in a simu-lated line-end mesoscale vortex within an MCS, but waslater exceeded by tilting of perturbation vorticity. How-ever, no one has definitively established that the cyclonicmember of a pair of line-end vortices and an MCV ofthe kind studied herein are precisely the same phenom-enon.

5. Comparison between two forms of the vorticitybudget

For the vorticity budget we used the conventionalform of the vorticity equation, (1), for two reasons. First,the conventional form has explicit terms for advection,divergence, and tilting, so one can easily grasp the phys-ical processes represented. Second, the great majorityof other vorticity budgets for observed MCVs are basedon the conventional form (e.g., Johnson and Bartels1992; Brandes and Ziegler 1993; Keenan and Rutledge1993; Scott and Rutledge 1995; Chong and Bousquet1999; Bousquet and Chong 2000), so it is easy to putour results in a larger context.

However, as explained in section 2d, the local ten-dency in the conventional form of the vorticity equationis very sensitive to the character of spatial derivativesbecause it is the sum of four large terms that often nearlycancel one another. To assess how much our primaryresults depend on our choice of vorticity equation, wetested a second budget using the divergence, or the flux,form of the vorticity equation, (4)–(6).

The two methods produced grossly similar resolvedtendencies, but with noteworthy differences in the de-tails (Fig. 12). In the lower 65% of the troposphere,below about 8 km AMSL, the two budgets agree well,although the divergence form tended to produce slightlylarger tendencies. In the upper troposphere, the diver-gence form produced significantly larger tendencies. Ina few layers, such as between 8 and 11 km AMSL at1500 UTC (Fig. 12b), the two budgets even producedtendencies of opposite sign. However, throughout muchof the troposphere, the signs of the vertical derivativeof tendency were virtually identical between the bud-gets, as were the altitudes of local extrema. Many ofthe most general conclusions one might draw from the


FIG. 12. Resolved part of the local tendency of relative vorticitycalculated according to the conventional (dashed) and divergence(solid) forms of the budget. Profiles are at (a) 1200, (b) 1500, and(c) 1800 UTC on 1 Aug 1996 for a 28 3 28 area centered on theMCV in the middle troposphere; the profiles have not been otherwiseaveraged or smoothed.

vertical structure of the resolved tendency, if not itsactual value at a given altitude, in our case seem to belargely independent of the form of the budget used.

There are at least two reasons for the differences be-tween the resolved tendencies from the divergence andconventional forms of the budget. First, the divergenceform of the vorticity equation, (4)–(6), has fewer terms

(and fewer derivatives) than the conventional form, (1),so there are fewer calculations. Second, and almost cer-tainly more crucial, the terms in the two budgets arenot calculated at an identical set of grid points. Becausethe divergence form involves a vector with no verticalcomponent, an areally averaged budget is calculatedsimply by applying (6) at the grid points along the pe-rimeter of the area in question. The divergence form isunaffected by vorticity extrema inside the perimeter.However, the conventional form of the budget is af-fected by extrema inside the perimeter because termsin the budget are calculated at every grid point, thenthe results are averaged.

Fortunately, the differences between the divergenceand conventional forms of the budget do not call intoquestion the primary results of this study. First, the me-soscale and synoptic components of wind contributedsignificant vorticity to the MCV of 1 August 1996; theformer mainly through convergence, vorticity, andthree-dimensional wind; the latter mainly through plan-etary vorticity and horizontal wind. Second, the twolargest net, resolved sources of vorticity were conver-gence and tilting, the prominence of which varied duringthe lifetime of the MCV. Mesoscale convergence of me-soscale and synoptic vorticity produced the single big-gest increase in the maturing MCV’s strength. Most ofthe tendency due to tilting was from mesoscale up- anddowndrafts acting on horizontal synoptic and mesoscalevorticity.

6. Synthesis

We presented a scale-discriminating vorticity budgetof a mesoscale convective vortex (MCV) generated bya mesoscale convective system (MCS). The MCS, whichcomprised a leading convective line and trailing strat-iform region, traversed Kansas and Oklahoma on 1 Au-gust 1996 and displayed many of the features in radarreflectivity common to systems that generate MCVs.

Because the MCS matured, decayed, and dissipatedin the National Oceanic and Atmospheric Administra-tion (NOAA) Wind Profiler Network (NPN), we wereable to analyze sources and sinks of vorticity in theMCV over 9 h on scales between those of semidailyoperational rawinsondes and Doppler radars. We useda Barnes bandpass filter to divide observed wind into acomponent that was predominantly synoptic back-ground wind and a component that was predominantlya mesoscale perturbation to that background wind.

The most important result from the vorticity budgetis that both the synoptic background wind and the me-soscale perturbation in wind contributed significant vor-ticity to the MCV of 1 August 1996. Vorticity tendencyfrom horizontal advection was due almost entirely toadvection of mesoscale vorticity by the synoptic wind.Tendency from vertical advection was due almost en-tirely to advection of mesoscale vorticity by the me-soscale wind. The largest contribution to vorticity ten-


dency from divergence was due to convergence of bothsynoptic and mesoscale vorticity by the mesoscale wind.Finally, only the horizontal variation of up- and down-drafts in the mesoscale wind contributed appreciably totilting horizontal vorticity, and this horizontal vorticitywas in the vertical shear of both the synoptic wind andthe mesoscale wind.

If the MCV of 1 August 1996 is representative, theresults herein suggest that completely realistic simula-tions of MCVs should include planetary vorticity anda plausible, three-dimensionally heterogeneous back-ground wind. However, an observational study such asthis does not clearly establish precisely in what waysan MCV is sensitive to heterogeneity in the environ-ment. Studies involving numerical simulations are bettersuited to that question.

Acknowledgments. The bulk of this research was con-ducted while the first author was at Colorado State Uni-versity and was supported by the National ScienceFoundation and the National Aeronautics and Space Ad-ministration under the respective Grants ATM 9618684and NCC5-288 SUPP 0002. The remainder, conductedwhile the first author was at the National Severe StormsLaboratory working under D. P. Jorgensen, was sup-ported through an associateship awarded by the NationalResearch Council.

Profiler data and code to read them are from S. B.Trier and C. F. Shih of the National Center for Atmo-spheric Research (NCAR); WSI’s NOWrad radar re-flectivity is from the Global Hydrology Resource Cen-ter; and GCIP/ESOP-96 data are from the Joint Officefor Scientific Support of the University Corporation forAtmospheric Research and the National Oceanic andAtmospheric Administration (NOAA).

Contributions from the following people improvedour research and writing: C. A. Davis and S. B. Trierof NCAR; J. E. Nachamkin of the Naval Research Lab-oratory; E. I. Tollerud of the Forecast Systems Labo-ratory; J. P. Kossin of the Space Science and Engi-neering Center; P. T. Haertel of the NOAA/AeronomyLaboratory; M. D. Parker of the University of Nebraska;P. C. Ciesielski, W. R. Cotton, M. T. Montgomery, andJ. A. Ramirez of Colorado State University (CSU); andthree anonymous reviewers. M. D. Parker, in particular,was extremely helpful. R. K. Taft of CSU provided in-valuable computer support.

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ol o archjohnson.atmos.colostate.edu/publications/refereed/...mcvs should include planetary...

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