-
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
-
784 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
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.
-
15 MARCH 2003 785K N I E V E L A N D J O H N S O N
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
-
786 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
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
-
15 MARCH 2003 787K N I E V E L A N D J O H N S O N
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
-
788 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
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.
-
15 MARCH 2003 789K N I E V E L A N D J O H N S O N
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-
-
790 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
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
-
15 MARCH 2003 791K N I E V E L A N D J O H N S O N
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
-
792 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
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-
-
15 MARCH 2003 793K N I E V E L A N D J O H N S O N
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.
REFERENCES
Barnes, S. L., 1973: Mesoscale objective analysis using
weightedtime-series observations. NOAA Tech. Memo. ERL NSSL-62,60
pp. [Available from National Severe Storms Laboratory, Nor-man, OK
73069.]
Bartels, D. L., and R. A. Maddox, 1991: Midlevel cyclonic
vorticesgenerated by mesoscale convective systems. Mon. Wea.
Rev.,119, 104–118.
Biggerstaff, M. I., and R. A. Houze Jr., 1991: Kinematic and
pre-
cipitation structure of the 10–11 June 1985 squall line.
Mon.Wea. Rev., 119, 3034–3065.
Bousquet, O., and M. Chong, 2000: The oceanic mesoscale
convec-tive system and associated mesovortex observed 12
December1992 during TOGA–COARE. Quart. J. Roy. Meteor. Soc.,
126,189–211.
Brandes, E. A., 1990: Evolution and structure of the 6–7 May
1985mesoscale convective system and associated vortex. Mon.
Wea.Rev., 118, 109–127; Corrigendum, 118, 990.
——, and C. L. Ziegler, 1993: Mesoscale downdraft influences
onvertical vorticity in a mature mesoscale convective system.
Mon.Wea. Rev., 121, 1337–1353.
Brock, F. V., K. C. Crawford, R. L. Elliott, G. W. Cuperus, S.
J.Stadler, H. L. Johnson, and M. D. Eilts, 1995: The
OklahomaMesonet: A technical overview. J. Atmos. Oceanic Technol.,
12,5–19.
Ceselski, B. F., and L. L. Sapp, 1975: Objective wind field
analysisusing line integrals. Mon. Wea. Rev., 103, 89–100.
Chen, S. S., and W. M. Frank, 1993: A numerical study of the
genesisof extratropical convective mesovortices. Part I: Evolution
anddynamics. J. Atmos. Sci., 50, 2401–2426.
Chong, M., and O. Bousquet, 1999: A mesovortex within a
near-equatorial mesoscale convective system during TOGA COARE.Mon.
Wea. Rev., 127, 1145–1156.
Cotton, W. R., M. S. Lin, R. L. McAnelly, and C. J. Tremback,
1989:A composite model of mesoscale convective complexes. Mon.Wea.
Rev., 117, 765–783.
Cram, T. A., M. T. Montgomery, and R. F. A. Hertenstein, 2002:
Earlyevolution of vertical vorticity in a numerically simulated
ide-alized convective line. J. Atmos. Sci., 59, 2113–2127.
Davis, C. A., and M. L. Weisman, 1994: Balanced dynamics of
me-soscale vortices produced in simulated convective systems.
J.Atmos. Sci., 51, 2005–2030.
Durran, D. R., and J. B. Klemp, 1982: On the effects of moisture
onthe Brunt–Väisälä frequency. J. Atmos. Sci., 39,
2152–2158.
Esbensen, S. K., 1993: Cumulus effects on vorticity. The
Represen-tation of Cumulus Convection in Numerical Models,
Meteor.Monogr., No. 46, Amer. Meteor. Soc., 93–98.
——, E. I. Tollerud, and J.-H. Chu, 1982: Cloud-cluster-scale
cir-culations and the vorticity budget of synoptic-scale waves
overthe eastern Atlantic intertropical convergence zone. Mon.
Wea.Rev., 110, 1677–1692.
Frank, W. M., 1983: The cumulus parameterization problem.
Mon.Wea. Rev., 111, 1859–1871.
Fritsch, J. M., J. D. Murphy, and J. S. Kain, 1994: Warm-core
vortexamplification over land. J. Atmos. Sci., 51, 1780–1807.
Gallus, W. A., Jr., and R. H. Johnson, 1992: The momentum
budgetof an intense midlatitude squall line. J. Atmos. Sci., 49,
422–450.
Hertenstein, R. F. A., and W. H. Schubert, 1991: Potential
vorticityanomalies associated with squall lines. Mon. Wea. Rev.,
119,1663–1672.
Houze, R. A., Jr., S. A. Rutledge, M. I. Biggerstaff, and B. F.
Smull,1989: Interpretation of Doppler weather radar displays of
mid-latitude mesoscale convective systems. Bull. Amer. Meteor.
Soc.,70, 608–619.
Johnson, R. H., and D. L. Bartels, 1992: Circulations associated
witha mature-to-decaying midlatitude mesoscale convective
system.Part II: Upper-level features. Mon. Wea. Rev., 120,
1301–1320.
Keenan, T. D., and S. A. Rutledge, 1993: Mesoscale
characteristicsof monsoonal convection and associated stratiform
precipitation.Mon. Wea. Rev., 121, 352–374.
Knievel, J. C., and R. H. Johnson, 2002: The kinematics of a
mid-latitude, continental mesoscale convective system and its
me-soscale vortex. Mon. Wea. Rev., 130, 1749–1770.
Koch, S. E., M. DesJardins, and P. J. Kocin, 1983: An
interactiveBarnes objective map analysis scheme for use with
satellite andconventional data. J. Climate Appl. Meteor., 22,
1487–1503.
Maddox, R. A., 1980: An objective technique for separating
mac-
-
794 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
roscale and mesoscale features in meteorological data. Mon.Wea.
Rev., 108, 1108–1121.
Mapes, B. E., 1993: Gregarious tropical convection. J. Atmos.
Sci.,50, 2026–2037.
Menard, R. D., and J. M. Fritsch, 1989: A mesoscale
convectivecomplex-generated inertially stable warm core vortex.
Mon.Wea. Rev., 117, 1237–1261.
Montgomery, M. T., and J. Enagonio, 1998: Tropical
cyclogenesisvia convectively forced vortex Rossby waves in a
three-dimen-sional quasigeostrophic model. J. Atmos. Sci., 55,
3176–3207.
O’Brien, J. J., 1970: Alternative solutions to the classical
verticalvelocity problem. J. Appl. Meteor., 9, 197–203.
Raymond, D. J., and H. Jiang, 1990: A theory for long-lived
me-soscale convective systems. J. Atmos. Sci., 47, 3067–3077.
Reed, R. J., and R. H. Johnson, 1974: A vorticity budget of
synopticwave disturbances in the tropical western Pacific. J.
Atmos. Sci.,31, 1784–1790.
Rogers, R. F., and J. M. Fritsch, 2001: Surface cyclogenesis
fromconvectively driven amplification of midlevel mesoscale
con-vective vortices. Mon. Wea. Rev., 129, 605–637.
Schubert, W. H., and J. J. Hack, 1982: Inertial stability and
tropicalcyclone development. J. Atmos. Sci., 39, 1687–1697.
——, ——, P. L. Silva Dias, and S. R. Fulton, 1980:
Geostrophicadjustment in an axisymmetric vortex. J. Atmos. Sci.,
37, 1464–1484.
Scott, J. D., and S. A. Rutledge, 1995: Doppler radar
observations
of an asymmetric MCS and associated vortex couplet. Mon.
Wea.Rev., 123, 3437–3457.
Skamarock, W. C., M. L. Weisman, and J. B. Klemp, 1994:
Three-dimensional evolution of simulated long-lived squall lines.
J.Atmos. Sci., 51, 2563–2584.
Sui, C.-H., and M. Yanai, 1986: Cumulus ensemble effects on
thelarge-scale vorticity and momentum fields of GATE. Part I:
Ob-servational evidence. J. Atmos. Sci., 43, 1618–1642;
Corrigen-dum, 46, 1630.
Tollerud, E. I., and S. K. Esbensen, 1983: An observational
study ofthe upper-tropospheric vorticity fields in GATE cloud
clusters.Mon. Wea. Rev., 111, 2161–2175.
Trier, S. B., C. A. Davis, and W. C. Skamarock, 2000:
Long-livedmesoconvective vortices and their environment. Part II:
Inducedthermodynamic destabilization in idealized simulations.
Mon.Wea. Rev., 128, 3396–3412.
Weisman, M. L., and C. A. Davis, 1998: Mechanisms for the
gen-eration of mesoscale vortices within quasi-linear convective
sys-tems. J. Atmos. Sci., 55, 2603–2622.
Zhang, D.-L., 1992: The formation of a cooling-induced
mesovortexin the trailing stratiform region of a midlatitude squall
line. Mon.Wea. Rev., 120, 2763–2785.
——, and J. M. Fritsch, 1988: A numerical investigation of a
con-vectively generated, inertially stable, extratropical
warm-coremesovortex over land. Part I: Structure and evolution.
Mon. Wea.Rev., 116, 2660–2687.