the genesis of severe, long-lived bow echoes

15 FEBRUARY 1993 WEISMAN 645

The Genesis of Severe, Long-Lived Bow Echoes

MORRIS L. WEISMAN

National Center for Atmospheric Research,* Boulder, Colorado

(Manuscript received 31 August 1991, in final form 26 May 1992)

ABSTRACT

A series of idealized simulations using a nonhydrostatic cloud model is used to investigate the genesis of bowechoes (a bow-shaped system of convective cells that is especially noted for producing long swaths of damagingsurface winds). It is hypothesized that severe, long-lived bow echoes represent a dynamically unique form ofmesoconvective organization' being produced for a restricted range of environmental conditions, including aconvective available potential energy (CAPE) of at least 2000 m2 s"2 and vertical wind shears of at least 20m s"1 over the lowest 2.5-5 km AGL. The key structural features include a 40-100-km-long bow-shaped segmentof convective cells, with a strong rear-inflow jet extending to the leading edge of the bow at 2-3 km AGL, andcyclonic and anticyclonic eddies (referred to as "bookend" vortices) on the northern and southern flanks ofthe bowed segment, respectively. This structure characteristically develops three to four hours into the lifetimeof a convective system and may remain coherent for several hours.

The evolution of this coherent structure occurs systematically as the convectively produced cold pool strengthensover time, eventually producing a circulation that overwhelms the ambient shear. This forces the convectivecells to advect rearward above the cold air and weaken. The horizontal buoyancy gradients along the back edgeof these rearward-advecting cells subsequently generate an elevated rear-inflow jet that extends to near theleading edge of the cold pool. The circulation of this jet helps negate the circulation of the cold pool, reestablishingdeep, forced lifting at the leading edge of the system. This elevated rear-inflow jet is also enhanced through thedevelopment of bookend vortices. Such vortices are produced at the ends of a convective line segment as vortexlines inherent in the ambient vertically sheared environment are first tilted upward by the convective updraftsand then tilted downward and stretched by the convective downdrafts. The development of these featuresrequires both large amounts of CAPE and strong vertical wind shear in the environment of these systems, as isconsistent with the observed environments of many severe, long-lived bow echoes.

1. IntroductionThe identification and understanding of convective

structures that are coherent in both space and timehave been ongoing goals of convective research. Thesestructures are intriguing not only due to their uniquedynamical character but also due to their enhancedpredictability. Many of these structures have been thesubject of previous inquiry, the supercell being a no-table example (e.g., KJemp 1987). In this paper, wewill discuss a long-lived mesoconvective structure thattakes the form of a 60-100-km long bow-shaped seg-ment of cells and that evolves either from an isolatedcell or as part of a larger-scale squall line. These "bowechoes" (Fujita 1978) have been identified on radarfor more than 30 years and are especially noted forproducing long swaths of damaging surface winds.However, the physical mechanisms responsible for theirstructure and longevity have yet to be explained. In

* The National Center for Atmospheric Research is sponsored bythe National Science Foundation.

Corresponding author address: Dr. Morris L. Weisman, NationalCenter for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000.

the following, we will investigate bow echoes througha series of idealized simulations using a nonhydrostaticnumerical cloud model. We will hypothesize, in par-ticular, that certain severe, long-lived bow echoes rep-resent a dynamically unique form of mesoconvectiveorganization that occurs for a limited set of environ-mental conditions: in particular, large amounts ofthermodynamic instability and strong low-level verticalwind shear.

The tendency for convective cells to organize intoor be organized by mesoscale systems has been knownfor quite some time. Perhaps the most intensely in-vestigated example is the squall line, which is observedto exist on scales ranging from 100 to over 1000 km(e.g., Houze and Hobbs 1982; Bluestein and Jain1985). More recently, investigators have focused on abroader range of long-lived mesoscale convective sys-tems (MCSs) (e.g., Maddox 1980; Houze et al. 1990),many of which begin as squall lines. In all of thesecases, the system is envisioned to be composed of asequence of shorter-lived, independent convective cellsthat contribute collectively to the much larger system-scale structure.

The first reference to a system-scale organization thatis larger than cell scale but smaller than squall-line or

1993 American Meteorological Society

646 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 50, No. 4

MCS scale was presented by Nolen (1959), who iden-tified a structure referred to as a line echo wave pattern(LEWP), which he denned to be" . . . a configurationof radar echoes in which a line of echoes has been sub-jected to an acceleration along one portion and/or adeceleration along that portion of the line immediatelyadjacent, with a resulting sinusoidal mesoscale wavepattern in the line." Hamilton (1970) further eluci-dated the significance of LEWPs by noting the asso-ciation of this feature with damaging straight-line windsand tornadoes. He also deduced that the echo bulgewas associated with an intense mesohigh, with the crestof the feature associated with a mesolow. Strong, dam-aging winds were attributed to the intense pressure gra-dient produced by these mesoscale pressure features.The term "bow echo" was first coined by Fujita (1978)in reference to the "bulging" echoes of Hamilton(1970): these echoes could occur individually or aspart of an LEWP as identified by Nolen (1959). BothFujita and, more recently, Przybylinski and Gery(1983) emphasize the association of bow echoes withlong swaths of damaging straight-line winds.

A typical morphology of radar echoes associated witha bow echo, as envisioned by Fujita (1978; referred toas a "downburst" in this reference), is presented inFig. 1. The system usually begins as a single, large, andstrong convective cell that may be relatively isolatedor may be part of a more extensive squall line. As thestrong surface winds develop, the initial cell evolvesinto a bow-shaped segment of cells, with the strongestwinds occurring at the apex of the bow. The cells atthe ends of the segment may appear to move rearwardrelative to the center of the bow. During the most in-tense phase of its life, the center of the bow may forma "spearhead," with cyclonic and anticyclonic motionof the cells on the left and right flanks of the bow. Astrong rear-inflow jet with its core at the apex of thebow is thought to be associated with the bulging andspeeding up of the radar echoes. This concentrated jetis proposed by Fujita to be the source of the damagingwinds. During the declining stage, the system oftenevolves into a comma-shaped echo with a cyclonically

rotating head. Many of these hypothesized flow featureshave recently been confirmed via Doppler radar ob-servations of bow echo-type systems (e.g., Burgess andSmull 1990; Schmidt and Cotton 1989). While bow-shaped convective systems are observed to occur overover a wide range of time and space scales, the moreorganized, severe bow echoes described above typicallyrange in size from about 40 to 120 km and often havelifetimes of several hours.

Even though the observations reveal an organizationto the bow echo that is larger-than-cell scale, a signif-icant number of the documented cases also suggest thatindividual supercells may be incorporated within thelarger-scale structure (e.g., Przybylinski and DeCaire1985; Schmidt and Cotton 1989; Johns and Leftwich1988; Smith 1990). However, the frequent observationof bow echoes without apparent supercells suggests thatsupercell processes are not crucial to their existence.

A recent study by Johns and Hirt (1987) offers someadditional insights into bow echoes through a clima-tological study of a related phenomenon, referred toas a "derecho." The term "derecho" is used to describeconvective systems that produce straight-line convec-tive wind gusts greater than 26 m s " 1 within a concen-trated area with a major axis length of at least 400 km.The gusts must also show a systematic pattern of pro-gression, with no more than 3 h elapsing between suc-cessive wind damage events. Such systems have beenobserved to have lifetimes of as long as 18 h, producinga swath of damaging winds hundreds of kilometers wideand 1000 km long. During the period of May-Augustfor the years 1980-83, Johns and Hirt identified 70derecho cases in the United States, most of which oc-curred in the upper regions of the Midwest.

In identifying derechos, investigators generally at-tempt to distinguish between those convective systemsthat produce continuous swaths of damaging windsand those that are associated with more isolated, short-lived wind events. Beyond this, there is no identificationof the type of convective structure producing the windevent. The damage swath produced by an isolated su-percell or a line of supercells (not including tornadic

LargeStrong Echo

Tall

BOW EchoCyclonic

Comma Echo

Rotating Mead

B

FIG . 1. A typical morphology of radar echoes associated with bow echoes that produce strongand extensive downbursts, labeled DB on the figure (from Fujita 1978).


events) may easily satisfy the criteria for a derechoevent. However, an examination of the radar-echoconfigurations associated with these systems stronglysuggests that the majority of cases involve bow echoesor LEWPs (Przybylinski and Decaire 1985; Johns andHirt 1987).

Johns and Hirt found that the most significant at-tributes of the derecho environment were the extremeamounts of convective instability and low-level mois-ture. Surface dewpoints were commonly greater than20°C, and the average lifted index was -9°C. In ad-dition, wind strengths in the low-to-midtropospherewere greater than that for other types of severe-weatheroutbreaks. The average 500-mb winds were estimatedto be 21 m s~', with the average 700-mb winds at 17m s"1. Also, over 80% of the events began along or tothe north of an east-west oriented quasi-stationarythermal boundary and then moved along the boundary.The significance of this association is as yet unclear,but certain possibilities include the enhanced low-levelconvergence along the zone for triggering convectiveevents, the enhanced instability that might be realizedowing to the deepening of the moist layer along thezone, and the enhanced low-level vertical wind shearthat would also be generated along such zones (i.e.,such generation can be explained with thermal wind-type arguments). These environmental conditions havebeen established further in a more recent climatologicalstudy by Johns et al. (1990), and are also consistentwith the environments observed for the individual bowecho cases referenced before.

The tendency for a convective cell to evolve into abow-shaped system of cells for certain environmentsis documented in several of the previous modelingstudies (e.g., Weisman and Klemp 1986). Fundamen-tally, an updraft produces rain that falls and evaporates,thereby producing a pool of cold air that spreads alongthe ground. This spreading cold pool produces con-vergence and lifting along its leading edge that can thentrigger new cells. However, rather than a cold pool pro-ducing a complete circle of new cells around the initialstorm, cells are favored along a bow-shaped arc orientedperpendicular to the vertical wind-shear vector. Theability to trigger new cells along this arc increases dra-matically as the amount of vertical wind shear in-creases, and also increases if the wind shear is confinedto the lowest 2-3 km AGL.

An explanation for this behavior is offered by Ro-tunno et al. (1988, hereafter RKW), who found thatthe best conditions for triggering cells along a spreadingcold pool occur when the horizontal vorticity generatedby the buoyancy gradient at the edge of the cold poolis matched by the opposing horizontal vorticity inher-ent in the ambient low-level vertical wind shear. In thissituation, a vertical jet of air is created at the leadingedge of the cold air that produces deeper lifting thanif the shear were not present. This mechanism may,on its own, explain the strength and longevity of many

of the observed convective systems that have been la-beled as bow echoes. However, more recent simulationssuggest that a unique dynamical structure may evolvewithin some of these systems that is not explained bythe previous studies.

An example of such a feature occurs in a squall-linesimulation previously presented by Weisman et al.(1988, hereafter WKR). The environmental conditionsfor this case include a convective available potentialenergy (CAPE) of 2400 m2 s~2 and a unidirectionalvertical wind shear of 25 m s"1 over the lowest 2.5 kmAGL, with the vertical wind-shear vector oriented per-pendicular to the squall line. The winds remain con-stant above 2.5 km. As described by WKR, the storm-induced cold pool remains well matched to the ambientshear throughout the simulation, with strong, erect cellscontinually being regenerated along the leading edgeof the cold air. By 220 min (Fig. 2a), a 50-km-longline segment of cells has developed in the southernportion of the domain, with cyclonic and anticycloniceddies apparent on the northern and southern ends ofthis line segment, respectively. More isolated cells arestill scattered along the remaining portions of the line.Over the next 40 min, a strong rear-inflow jet developsbehind the center of this line segment, and by 260 min,the rear inflow has intensified to greater than 20 m s~'over the ambient flow. This structure is depicted at 300min (Fig. 2b) and continues in quasi-steady fashionfor an additional hour. A smaller bowed segment hasdeveloped north of the original feature, but does notretain its structure for as long a period of time.

The feature described in the above simulation rep-resents a form of mesoconvective organization that hasyet to be identified or investigated in numerical studies.The similarity between this feature and the availableobservations of bow echoes during the early part oftheir evolution (e.g., up to the comma-echo stage inFig. 1) is intriguing. Common features include its bowshape and size, long life, intense rear-inflow jet withvortices on either side, and strong flow at the surface.In addition, both develop either from an isolated cellor as part of a squall line, and both occur for similarenvironmental conditions, consisting of large amountsof thermodynamic instability and strong low-level ver-tical wind shear. Finally, both are generated severalhours into the life cycle of the convective system. It isthus hypothesized that the model-produced featuremay be representative of many of the observed long-lived convective structures that have been labeled asbow echoes.

The goal of this paper is to document the physicalprocesses that develop and maintain these long-livedmesoconvective structures. This will be accomplishedthrough the analysis of idealized simulations that re-produce the bow-echo structure using the simplestpossible set of initial conditions. The discussion willconcentrate particularly on one simulation that tracesthe evolution of an isolated cell in a horizontally ho-


mogeneous environment with unidirectional verticalwind shear, using the same thermodynamic and windprofiles specified in the squall-line example discussedbefore. This simulation reproduces all the major fea-tures of the squall-line bow echo described in Fig. 2,including the midlevel vortices at the ends of convectivesegment and the strong, elevated rear-inflow jet. Theanalysis of these features, however, is greatly simplified

SQUALL-LINE BOW ECHO150

i i i n n l i i / n n i i i i i M i i i i m i i r

165

165

X (km)

F I G . 2. Horizontal cross sections of system-relative flow and rain-water contours at 2.5 km AGL at (a) 220 min and (b) 300 min forthe squall line simulation described by WKR, using an environmentalvertical wind shear of 25 m s~' over the lowest 2.5 km AGL. Adomain speed of U = 22.5 m s"1 has been subtracted from the flowvectors. Vectors are plotted at every other grid point (a distance oftwo grid lengths represents 25 m s"1). The rainwater field is contouredusing a 2 g kg""' interval. Only a 150 km X 150 km portion of theentire domain is shown.

by tracing the evolution of a single cell rather than theevolution of a complete squall line. This simulationwill be referred to as the "idealized" bow echo.

The discussion begins in section 2 with a brief de-scription of the numerical model used for this study,followed in section 3 by a detailed description of theevolution of the idealized bow echo. Section 4 presentsan analysis of the rear-inflow jet based on the recentstudy of Weisman (1992), who considers the evolutionof two-dimensional circulations via the horizontal vor-ticity equation. This analysis explains much of the rear-inflow characteristics along the center of the bow echo,but must be expanded in section 5 to consider the in-fluence of the midlevel vortices that exist at the endsof the bowed system. In section 6, a set of simulationsis presented that tests the sensitivity of the results tothe details of the thermodynamic profile and the depthof the shear layer and helps clarify the environmentalconditions most conducive to bow echo genesis. Fi-nally, in section 7, the main findings of this study aresummarized and discussed in the context of future re-search initiatives.

2. Model formulation and experimental design

The simulations to be described in this study wereall completed using the three-dimensional nonhydro-static numerical cloud model described by Klemp andWilhelmson (1978), as also used by Weisman (1992).The model does not currently include ice processes orradiative heating. Open boundary conditions are em-ployed on the lateral boundaries, and a gravity-waveradiation condition is employed at the upper boundary(Klemp and Durran 1983). Free-slip conditions arespecified at the surface, with the vertical velocity set tozero.

Convective cells are initialized in a horizontally ho-mogeneous environment that is characterized by spec-ified vertical profiles of temperature, water vapor, andwind. In addition, the wind profiles are defined to beunidirectional, and the Coriolis force is set to zero.These conditions ensure a mirror-image symmetry tothe solution about an axis directed parallel to the ver-tical wind-shear vector. Thus, only half of the full do-main need be included. The effects of the Coriolis forceand directionally varying vertical wind shears on systemstructure will be presented in a future study. Stormsare initialized by placing an ellipsoidal warm thermalof 10-km horizontal and 1400-m vertical radius on thecenter of the symmetry axis. The thermal is given amaximum magnitude of 2°C at the center, which de-creases to zero at the edge, and is balanced hydrostat-ically with the pressure field. Flow begins as air accel-erates horizontally into the region of lowered pressurebeneath the thermal, with the resultant convergencecreating upward motion that releases the convectiveinstability.


The simulations to be described all use a horizontalgrid resolution of 2 km and a vertical grid resolutionof 750 m. The domain size is 80 km in the y, or cross-wind, direction; 160 km in the x, or alongwind, direc-tion; and 17.5 km in the z, or vertical, direction. Thesymmetry axis is located along y = 0, and thus, theeffective domain size in the y direction is also 160 km.Each simulation is run for 240 min. Several experi-ments were run to test the sensitivity of the results tothis model formulation. In one experiment, the hori-zontal resolution was increased to 1 km, keeping thedomain size the same. In another case, the vertical res-olution was increased to 350 m at low levels, stretchinguniformly to about 1 km at the top of the domain. Ina third experiment, the original grid resolution was re-tained, but the domain was increased in size to 160km along the y axis (effectively 320 km, including themirror image) by 640 km along the x axis to test theeffects of the boundaries. However, the present resultswere not significantly affected by any of these changes.

The environmental conditions used for the idealizedbow echo simulation are identical to those used in thesquall-line simulation discussed in the Introduction.The thermodynamic profile is characterized by a CAPE

IDEALIZED BOW ECHO

- 8 0 - 6 0 - 4 0

- 2 0 0 20

TEMPERATURE (°C)

FIG. 3. Skew T-Xogp diagram depicting the range of thermody-namic profiles used for the numerical simulations. The three tem-perature profiles are depicted by the thick solid lines, while the mois-ture profile is depicted by the thick dashed line. The temperatureprofile used for the idealized bow echo simulation is denoted by astar. The thick dotted line represents the parcel ascent curve for the14 g kg"1 surface mixing ratio cases included in Table 1.

30 -

g 2 0 -

\0 -1 A

- 3 0

-20 ;

- 10

60 120 180

TIME (min)

240

F I G . 4. Time series of maximum (solid) and minimum (dashed)vertical velocities (m s"') observed during the 240-min idealized bowecho simulation.

of 2377 m2 s 2 with relatively moist conditionsthroughout the troposphere (Fig. 3). Air that ascendsfrom near the surface experiences a small amount ofnegative buoyancy before reaching the level of freeconvection (LFC) at 1.7 km AGL (820 mb). Thisthermodynamic profile was also used in the sensitivitystudies of Weisman and Klemp (1982, 1984, 1986)and of WKR and RKW. Also included in Fig. 3 areadditional temperature profiles used for the environ-mental sensitivity experiments to be described in sec-tion 6. A unidirectional wind profile is specified alongthe y axis, with the wind increasing linearly from zeroat the surface to a maximum of Us = 25 m s~' at 2.5km AGL, with constant winds above. The sensitivityof the results to variations in this wind profile will alsobe discussed in section 6.

3. An idealized bow echo

a. General system characteristics

An overview of the evolution of the idealized bowecho is presented through a time series of the maximumand minimum vertical velocities (Fig. 4), along withhorizontal cross sections depicting updraft and down-draft locations, rainwater concentration, and approx-imate system-relative flow at the surface and 2.5 kmAGL at hourly intervals through 240 min (Figs. 5-8).The 2.5-km level was chosen as it clearly displays theevolution of the midlevel vortices and rear-inflow jet,which appear to be the distinguishing components ofthe bow-echo structure. Also, at this level, the system-relative ambient flow is close to zero, thereby accen-tuating the flow features generated by the convectivesystem. The vertical structure of the system is sum-marized in Fig. 9 through vertical cross sections ofrainwater concentration, equivalent potential temper-ature (8e), and system-relative flow along the symmetryaxis of the simulation domain, also at hourly intervals.


IDEALIZED BOW ECHOT = 60Min

(o)Z = 2.5 km-£ -60£ 30

— crTLL-(b) Z = 0km -

110

X(km)

FIG. 5. Horizontal cross sections of rainwater concentration andflow vectors at (a) 2.5 km AGL and (b) 0 km AGL at 60 min forthe idealized bow echo simulation. Vectors are presented at everyother grid point, with a vector length of two grid intervals being equalto a magnitude of 25 m s"'. A domain speed of U = 22.5 m s~' issubtracted from the flow field. The ground-relative system propagationspeed at this time is 18 m s"1. The rainwater is contoured at 2 g kg"'intervals, and the thick line represents the location of the gust frontat the surface. For (a), regions of updraft greater than 4 m s"' areshaded and regions of downdraft less than - 2 m s"' are stippled.The y = 0 boundary represents the axis of symmetry for the simu-lation, with only a 60 km X 80 km portion of the full domain pre-sented.

The initial evolution follows that described in theprevious modeling studies of Weisman and Klemp(1982, 1984, 1986) and WKR. The updraft triggeredalong the symmetry axis reaches a maximum strengthof 30 m s"1 at about 30 min into the simulation (Fig.4). This produces rainfall and evaporative cooling, re-sulting in the development of a downdraft and, sub-sequently, a cold pool that begins to spread along thesurface. By 60 min, the surface flow has begun to di-verge in response to the spreading cold air (Fig. 5b),with the downdraft intensifying to - 9 m s"1 (Fig. 4).The original rain cell is still on the symmetry axis, nearthe leading edge of the gust front, but the updraft has

split into a right- and left-moving pair in response tothe strong vertical wind shear (Fig. 5a; note that onlythe right mover is shown). The vertical cross sectionat this time cuts through the original rain cell, display-ing its essentially vertical orientation, with the remnantsof the original updraft still evident near its top (Fig.9a). As also described in RKW and WKR, the down-draft is fed from the front of the system, having orig-inated at midlevels where 8e is low, and spreads pri-marily rearward along the surface. The system prop-agation speed at this time (defined by the motion ofthe gust front relative to the ground) is 18 m s~'.

The split updraft strengthens to 30 m s"1 by 85 min(Fig. 4), but weakens thereafter and becomes incor-porated into a series of new cells that has developedalong the gust front between the original splitting pair.By 120 min, the system has evolved into two 30-km-long bow-shaped system of cells, extending to both sidesof the symmetry axis (Fig. 6), with updraft maximareduced to 23 m s"1 and downdraft minima increasedto —11 m s~' (Fig. 4). The surface flow diverges 12km behind the leading edge of these bow features, with

IDEALIZED BOW ECHOT = 120 Min

- 6 0

X(km)

FIG. 6. Same as Fig. 5 but at 120 min. The ground-relativesystem propagation speed at this time is 22.5 m s"1.


IDEALIZED BOW ECHOT =180 Min

E - 6 0

X(km)

FJG. 7. Same as Fig. 5 but at 180 min. The ground-relativesystem propagation speed at this time is 27 m s"1.

the ground-relative winds reaching speeds greater than30 m s"' (Fig. 6b). The remnants of the original splitupdraft at the southern end of the line segment arecorrelated with cyclonic vorticity at 2.5 km, but a largeanticyclonic eddy is now also evident behind this up-draft zone and is coincident with the downdraft andrain region. A mirror-image anticyclonic updraft andcyclonic downdraft exists at the northern end of thesystem, north of the symmetry axis (not shown). Thisvorticity is created as the vortex lines inherent in theambient shear are tilted into the vertical by the updraftand then are tilted back down and stretched in thedowndraft. This process is discussed in detail in section5. The downdraft vortices at the ends of the convectiveline segment remain prominent throughout the sim-ulation and are subsequently referred to as bookendvortices. A rear-inflow current has developed betweenthe bookend vortices, with a strength of nearly 10 ms"1

close to each vortex (relative to the ambient 2.5-kmflow; i.e., subtract 2.5 m s~' from the vector magnitudespresented in the figures), but weakening to 3 m s"1

along the symmetry axis.

The vertical cross section along the symmetry axisat 120 min (Fig. 9b) passes through a newly developingcell, with a plume of high 6e air from near the surfaceextending through the entire vertical extent of thestorm. This updraft current still spreads predominantlydownshear aloft. The downdraft at this location is notvery strong, but low 6e air from midlevels has begunto descend from the rear of the system. The systempropagation speed at this time has increased to 22.5m s"1.

Between 120 and 180 min, the system continues toexpand in size, with the rear inflow strengthening con-siderably along the symmetry axis (now 15 m s"1 rel-ative to the ambient 2.5-km flow) and extending rear-ward nearly 40 km from the leading edge of the system(Fig. 7). It also expands in the vertical, reaching a depthof 3.5 km (Fig. 9c). The core of the jet remains atabout 2.5 km AGL as it penetrates to near the leadingedge of the gust front before being diverted upwardand downward as it collides with the updraft current.Potentially cold midlevel air that supplies the down-draft and surface outflow circulation now feeds entirely

IDEALIZED BOW ECHOT = 240 Min

126

X(km)

FIG. 8. Same as Fig. 5 but at 240 min. The ground-relativesystem propagation speed at this time is 29 m s"'.


IDEALIZED BOW ECHO

(a) 6 0 Min

I : : : z z.~ ^ . - - - i .

X (km)

FIG. 9. Vertical cross sections of Se, flow vectors and rainwaterconcentration along the y = 0 boundary at (a) 60 min, (b) 120 min,(c) 180 min, and (d) 240 min for the idealized bow-echo simulation.Vectors are presented at every other grid point, with a vector lengthof two grid intervals being equal to a magnitude of 25 m s"'. Adomain speed of U = 22.5 m s~' is subtracted from the flow field.The ground-relative system propagation speed is (a) 18 m s"1, (b)22.5 m s~\ (c) 27 m s~\ and (d) 29 m s"'. Regions of $„ greaterthan 334 K are darkly shaded, and regions less than 326 K are lightlyshaded. Thick lines represent the 0 and 4 g kg"1 rainwater contours.Only an 80 km X 12 km portion of the domain is presented.

from the rear of the system, in association with therear-inflow jet, with the cold pool increasing in depthto nearly 3 km at its leading edge. The updraft at theleading edge of the system is still vertically orientedthrough 5.5 km, but is much stronger in the lower levels

than earlier. Above 5.5 km, however, most of the up-draft current turns abruptly rearward (upshear), witha smaller proportion continuing on a vertical trajectoryand exiting downshear at about 9 km. The surface flownow diverges just behind the leading edge of the gustfront, with the strongest ground-relative winds confinedto this narrow zone (Fig. 7b). The bookend vorticesare still evident at the ends of the convective line seg-ment, but have not changed in intensity during thisperiod. The system propagation speed has further in-creased to 27 m s~', with updraft maxima remainingbetween 20 and 22 m s"1 and downdraft minimaweakening to about - 8 m s " 1 (Fig. 4).

Figure 10 presents a series of air parcel trajectoriesthat are representative of the flow during this phase ofsystem development. Each trajectory is calculated usingmodel data saved every 5 min, and originates at 180min at the location indicated by a small dot. Air parcelsthat originate near the surface ahead of the system arelifted abruptly as they encounter the leading edge ofthe gust front. Parcels near the symmetry axis are liftedsteeply through about 6 km and then proceed upwardmore gradually as they advect rapidly rearward relativeto the leading edge of the system. Just away from thiscore region, parcels are not lifted as high initially butcontinue upward behind the system. The deepest as-cent, however, occurs in the more isolated cells outsideof the zone defined by the bookend vortices. Air parcelswithin the rear-inflow jet descend gradually from theirorigin at 3-4 km as they approach the leading edge ofthe system, with the lower portion of the jet then de-scending more abruptly and exiting rearward along thesurface. The upper portion of the jet, however, divertsupward within the updraft and exits to the rear of thesystem at about 8-10 km. Since this rear-inflow air ispotentially cold and convectively stable upon ascent,this suggests that this portion of the updraft must bestrongly forced.

The rear inflow continues to strengthen and extendrearward over the next 30-60 min, reaching magni-

T » 180 min

Z(km)

FIG. 10. Representative parcel trajectories for the idealized bow-echo simulation at 180 min. Dots represent the point of origin ofeach trajectory, with the trajectory integrated forward and backwardin time from that point. The surface gust front at / = 180 min isdenoted by the dashed line.


tudes greater than 20 m s"1 over much of the core re-gion of the system, but the basic configuration thatevolved by 180 min, consisting of the bookend vortices,elevated rear-inflow jet, and strong erect updraftthrough 6 km along the leading edge, remains essen-tially intact. The most significant change during thisperiod is a narrowing of the distance between thebookend vortices between 180 and 240 min (Figs. 7aand 8a). This represents an intriguing aspect of theevolution of these features, but its explanation is be-yond the scope of this study. In addition, the systempropagation speed increases to 29 m s"1. By 240 min,the rain field is in the form of a 50-km continuous bowextending between the bookend vortices, with moreisolated, strong cells extending 25 km on either side ofthis primary feature (Fig. 8). The vertical cross section(Fig. 9d) depicts an even stronger and deeper rear-inflow current than earlier, with the lifting at the leadingedge of the system also stronger and deeper. This basicstructure is maintained for an additional 90 min (notshown), with the system propagation speed averagingabout 27 m s"1.

b. Diagnostic pressure analysis

An analysis of the pressure field at 180 min offersfurther insights into the forces creating and maintainingthe intense circulation. For this purpose, the pressureis decomposed via a diagnostic pressure equation foranelastic Boussinesq flow, as described by Schlesinger(1980) and Rotunno and Klemp (1982, 1985):

V • (Cpp 6VVIT) = - V - ( p v Vv) +dB

where B represents the buoyancy, defined as

( i )

(2)

8 represents the potential temperature; qv, qc, and qr

represent the mixing ratio of water vapor, cloud water,and rainwater, respectively; and ir represents the Exnerfunction, defined as

(3)

Bars over individual variables refer to the initial un-disturbed state, which is a function of z only, whileprimes refer to perturbations from this initial state. Thefirst term on the rhs of (1) represents the contributionsto the pressure field due to variations in the velocityfield. The component of pressure derived from thisterm is often referred to as the dynamic pressure, irdn.The expansion of this term (not shown) demonstrateshow the dynamic pressure can be decomposed furtherinto terms representing fluid extension and fluid shear.The fluid shear terms represent the rotational com-ponent of the flow and contribute to negative pertur-

bations to the pressure field. The fluid extension termsare associated with converging and diverging flow andcontribute to positive pressure perturbations.

The second term on the rhs of (1) represents con-tributions to the pressure field due to vertical variationsin the buoyancy field. The component of pressure thusdiagnosed is often referred to as the buoyancy pressure,irb. These pressure perturbations are similar to thosethat would be derived by calculating the hydrostaticcomponent of the pressure field, with negative pressureperturbations occurring beneath positive buoyancyperturbations, and vice versa. However, irb representsa more accurate portrayal of the resultant pressure fieldsince it considers the full three-dimensional effects ofthe buoyancy perturbations on the fluid.

A vertical cross section of the pressure perturbationsand system-relative flow along the symmetry axis at180 min is presented in Fig. 11. The pressure field ishighlighted by a mesohigh near the surface, with amaximum magnitude of 3.5 mb behind the leadingedge of the gust front, and a mesolow at midlevels,with a maximum magnitude of —1.5 mb at 3.5 kmAGL, slightly to the rear of the surface mesohigh. Therear inflow accelerates in response to the negative hor-izontal pressure gradient associated with this mesolow,but decelerates and diverges vertically as it encountersa strong positive pressure gradient at the leading edgeof the system. The decomposition of the pressure fieldreveals that the buoyancy pressure is the dominantcontributor to both the mesolow and mesohigh. Thedynamic terms contribute significantly only along theleading edge of the surface gust front, where the flowis strongly convergent.

A plan view of the pressure perturbations and sys-tem-relative flow at 180 min at 2.5 km AGL is pre-sented in Fig. 12. The flow characteristics are similarto that portrayed in the vertical cross section, with therear inflow accelerating into a midlevel mesolow anddecelerating at the leading edge of the system. Thepressure decomposition again suggests that the buoy-ancy terms are the dominant contributor to the pressurefield. However, the dynamic terms off of the axis ofsymmetry are now seen to contribute significantly be-hind much of the leading edge of the gust front andespecially in the region of the bookend vortex. Thisdynamic lowering of pressure is associated with thestrong curvature of the flow field, which is due to thestrong vertical circulation behind the gust front as wellas the horizontal circulation associated with the book-end vortex. An additional vertical cross section takencloser to the bookend vortex (not shown) displays apressure pattern similar to that depicted along thesymmetry axis, with the exception that a 0.5-1-mb dy-namic pressure low is evident behind the gust front,extending upward to 6 km AGL.

The forces accelerating the major flow componentsof the idealized bow echo can be traced more quanti-tatively through a Lagrangian analysis of the u com-


T» 180 mina) VECTORS, P'

Z (km)

12c)

-0DYNAMIC

\<

0

D

PRESSURE

o 0/ :

Jlk ;30 110

X(km)

FIG. 11. Vertical cross section along the y = 0 axis of (a) pressureperturbations and flow vectors, (b) buoyancy contributions to thepressure field, and (c) dynamic contributions to the pressure field at180 min for the idealized bow-echo simulation. The pressure field iscontoured at 0.5-mb intervals, with the dashed and solid lines con-touring negative and positive perturbations, respectively. The vectorsare presented at every other grid point, with a vector length of twogrid intervals being equal to a vector magnitude of 25 m s"'. A domainspeed of V = 22.5 m s"1 is subtracted from the flow field. The ground-relative system propagation speed at this time is 27 m s"1.

ponent of the equation of motion, which may be writ-ten to display the contributions to u momentum fromthe dynamic and buoyancy pressure gradients sepa-rately (also ignoring mixing effects):

dt " " dx " " dx '

The individual contributions are determined by inter-polating the terms along parcel trajectories feeding intothe rear inflow and integrating the appropriate pressuregradients along the trajectories over time:

Udn = -dx

U=

(5)

(6)

Udn + Ub, (7)

= \ -Cp6v^dt,JT OX

T" 180 mina) VECTORS, P'

Z- 2 .5 km

- 6 0

b) BUOYANCY PRESSURE

Y(kml

c) DYNAMIC PRESSURE

X(krn)

FIG. 12. Horizontal cross section at 2.5 km AGL of (a) pressureperturbations and flow vectors, (b) buoyancy contributions to thepressure field, and (c) dynamic contributions to the pressure field at180 min for the idealized bow echo simulation. The pressure field iscontoured at 0.5-mb intervals, with the dashed and solid lines con-touring negative and positive perturbations, respectively. The vectorsare presented at every other grid point, with a vector length of twogrid intervals equal to a vector magnitude of 25 m s " ' . A domainspeed of u = 22.5 m s"1 is subtracted from the flow field. The ground-relative system propagation speed at this time is 27 m s"1. For (a),the labels A and B denote the starting point for trajectories discussedin Fig. 13.


where UQ is the value of the u component of the windat the start of the trajectory. The trajectories are cal-culated using gridded velocity data saved every 5 minduring the numerical simulation.

Of particular interest is an analysis of the forces re-sponsible for the generation of the strong rear-inflowjet. The two trajectories presented in Fig. 13 are rep-resentative of the range of results. Each trajectory orig-inates at 2.5 km AGL, as located on Fig. 13a, and isfollowed backward in time for 60 min. The path of thefirst trajectory (trajectory A) corresponds closely to thevertical cross section along the symmetry axis presented

a) Rear Inflow Trajectories

12

Z (km)

120 180

c)

U,

20

Trajectory

= 4 ms"1

B

U (msH)

15-

10-

5-

• 1 -*

150

/

1

180

TIME (min)

FIG . 13. Representative trajectories (a) feeding into the rear inflowat 180 min for the idealized bow echo simulation. The starting pointsfor trajectories A and B are also indicated in Fig. 12a. The surfacegust front at * = 180 min is denoted by the dashed line. The integrationof the momentum forcing terms along trajectories A and B are pre-sented in (b) and (c), respectively. The solid lines represent the actualmagnitude of U interpolated along the trajectqry, while the dashedand dotted lines represent the integrated contributions to U from thebuoyancy terms (UB) and dynamic terms (6U), respectively, as de-scribed in the text.

in Fig. 9 and represents much of the core region of thebow echo. The air parcel descends from a height of 3.5km over the 60-min period. The integrations of thebuoyancy and dynamic forcing terms along this tra-jectory are presented in Fig. 13b and suggest that thebuoyancy terms contribute almost two-thirds of theresultant velocity. The horizontal pressure gradient as-sociated with this term is produced almost entirely bythe distribution of buoyancy within the vertical crosssections perpendicular to the bow echo. As is discussedlater this portion of the rear inflow may thus be largelyinterpreted through the analysis of the processes thatredistribute the buoyancy field in this two-dimensionalvertical plane.

The second trajectory (trajectory B) is taken closeto the bookend vortex. For this trajectory, the buoyancyand dynamic contributions are nearly equal (Fig. 13c).As discussed before, part of these dynamical contri-butions are associated with the bookend vortices. Ananalysis of the development of the rear inflow withinthis portion of the system would thus also entail ananalysis of the processes producing the bookend vortex.

A similar analysis has also been completed to di-agnose the forces maintaining the steady updraft alongthe leading edge of the bow echo. These results (notshown) reveal that much of the forcing for this updraftis due to the dynamically induced upward-directedvertical pressure gradient that is evident along the lead-ing edge of the system in Fig. lie. Buoyancy effectscontributed only 20%-50% of the resultant updraftspeeds. This result parallels a similar conclusion forsupercell updrafts, whereby 60% of the updraft speedwas attributable to dynamically induced vertical pres-sure gradients (Weisman and Klemp 1984). In the caseof supercells, however, the upward-directed verticalpressure gradient is induced by a lowering of the pres-sure at midlevels associated with a strongly rotatingupdraft. For this idealized bow echo, the upward-di-rected vertical pressure gradient is created by the col-lision between the rear-inflow jet and the low-level in-flow from ahead of the system. In both cases, the dy-namic forcing produces a steadier circulation than ifbuoyancy effects were the only major contributor toupdraft production.

c. Summary

During the initial two hours of this simulation, theevolution of the idealized bow echo is dominated bythe splitting of the initial convective cell, as previouslydescribed, for example, by Klemp and Wilhelmson(1978b). During this period, the system-scale flow atmidlevels is dominated by the development of circu-lations in a horizontal plane that are associated withthe development of the bookend vortices. Associatedwith the development of the bookend vortices is a 5-10 ms"1 rear inflow that is strongest near the edge ofthe vortices. These bookend vortices remain a prom-


(a)

777777777///////////////////////////////////////

(b)

777/7///////////////////////////////////////////

(C)

//////7T/77//7//////////////////////////////////

(d)

FIG. 14. Four stages in the evolution of an idealized bow echo,(a) An initial updraft leans downshear in response to the ambientvertical wind shear, which is shown on the right, (b) The circulationgenerated by the storm-induced cold pool balances the ambient shear,and the system becomes upright, (c) The cold pool circulation over-whelms the ambient shear, and the system tilts upshear, producinga rear-inflow jet. (d) A new steady state is achieved whereby thecirculation of the cold pool is balanced by both the ambient verticalwind shear and the elevated rear-inflow jet. The updraft current isdenoted by the thick double-lined flow vector, with the rear-inflowcurrent in (c) and (d) denoted by the thick dashed vector. The shadingdenotes the surface cold pool. The thin, circular arrows depict themost significant sources of horizontal vorticity, which are either as-

inent feature of the midlevel flow through the remain-der of the simulation.

As the system expands, however, a stronger verticalcirculation develops in a broad zone between thebookend vortices, associated with the development ofa quasi-two-dimensional updraft and elevated rear-in-flow jet. As is demonstrated in the pressure analysis,the horizontal pressure gradient accelerating this rearinflow is primarily associated with a quasi-two-dimen-sional buoyancy-produced mesolow that extends aboveand behind the spreading cold pool. It is the develop-ment of this elevated rear inflow, which reaches inten-sities of greater than 20 m s"' along the symmetry axis,that appears to be the dominant factor promoting thenew steady-state structure by 180 min.

In light of this, we will proceed first with an analysisof the two-dimensional processes important to the de-velopment of this elevated rear-inflow jet and will ex-plain how it contributes to producing an intense, steadyuplift along the leading edge of the system. We willthen discuss in more detail the processes promotingthe development of the bookend vortices and will at-tempt to clarify their role in initiating and strengtheningthis rear-inflow circulation.

4. The development and role of the rear-inflow jet

A detailed analysis of the physical mechanisms re-sponsible for the development of the rear-inflow jetwithin this simulation has recently been presented ina companion study by Weisman (1992). In this study,which extends the recent work of RKW and Laforeand Moncrieff (1989), the generation of the rear inflowis analyzed via the two-dimensional horizontal vorticityequation for inviscid Boussinesq flow; for example,

dt dx'(8)

where 77 = du/dz — dw/dx and where B represents thebuoyancy, defined by (2). Within this framework, theonly source of horizontal vorticity is horizontal gra-dients of buoyancy. Thus, the analysis of the devel-opment of circulation is simplified to understandingthe evolution of the buoyancy field.

This perspective allows for a simple interpretationof the evolution of quasi-two-dimensional convectivesystems, based on the interactions between the buoyantupdraft, the storm-induced cold pool, and the ambientvertical wind shear. This evolution is presented sche-matically in Fig. 14 for an isolated cell evolving in avertically sheared environmental flow. Initially, theconvective cell leans downshear in response to the am-

sociated with the ambient shear or are generated within the convectivesystem, as described in the text. Regions of lighter or heavier rainfallare indicated by the more sparsely or densely packed vertical lines,respectively. The scalloped line denotes the outline of the cloud(adapted from Weisman 1992).


bient vertical wind shear (Fig. 14a). As a cold pooldevelops beneath the convection, the horizontal buoy-ancy gradients along the edge of the cold pool generatecirculation that, along the downshear edge of the coldpool, is of the opposite sense as the circulation inherentin the ambient shear. When this cold pool circulationbalances the circulation inherent in the ambient shear,deeper lifting is produced at the cold pool edge, re-sulting in stronger, more upright convective cells (Fig.14b). As the cold pool circulation continues tostrengthen, it eventually overwhelms the ambient shear,and the convective circulation begins to tilt rearwardover the cold air (Fig. 14c).

It is during this upshear-tilting phase of the systemthat a significant rear-inflow jet is characteristicallygenerated. This occurs as the horizontal buoyancy gra-dients along the rear edge of the buoyant plume aloftand cold pool near the surface generate horizontal vor-ticity, thereby accelerating the flow from rear to frontat midlevels (as indicated in Fig. 14c by the bold arrowbetween the positive and negative circulation coupletat the back edge of the system). This mechanism isequivalent to that described via the diagnostic pressureanalysis (section 3c) in that this configuration of thebuoyancy field also implies a minimum in the buoy-ancy-derived pressure field at midlevels that acceleratesthe rear-inflow current.

For the idealized bow echo, the transition to an up-shear-tilted circulation occurs shortly after 120 min.Over the next 60 min, the rear inflow along and nearthe symmetry axis accelerates from 5 to 20 m s~'. Asdescribed by RKW, this upshear-tilting phase often de-notes the beginning of the decay stage of the convectivesystem, as the lifting along the leading edge of the gustfront becomes shallower than during the earlier phasesof system evolution (due to the dominating influenceof the cold pool). However, this tendency toward decaycan be superseded by the development of an elevatedrear-inflow jet, as is presented schematically in Fig. 14d(Weisman 1992). In this scenario, the rear-inflow jetis characterized by the opposite sense of horizontalvorticity beneath the jet level as is generated by thecold pool, and may thereby counteract some of thenegative influence of the cold pool circulation. For theidealized bow echo, the strength of the horizontal vor-ticity beneath this elevated jet is sufficient to again pro-mote deep lifting at the leading edge of the gust front,leading to the continual regeneration of strong erectupdrafts at that location. It is this forced lifting, pro-moted by the development of the strong, elevated rear-inflow jet, that appears to be the primary factor pro-moting the strength and longevity of this idealized bowecho.

Once the updraft air is lifted above the level of therear-inflow jet, it is drawn rapidly rearward above thecold pool under the influence of the horizontal vorticityassociated with the upper portion of the jet. Thus, whilethe low-to-midlevel updrafts during this phase in the

systems life are actually stronger than at earlier times,the updraft at higher levels is weaker, and the convec-tive cells shallower. This explains why the updraft tra-jectories at the core of the bow echo in Fig. 10 areshallower than for the more isolated cells along theperiphery of the system.

As described by Weisman (1992), the environmentalconditions conducive to the development of strong,elevated rear-inflow jets include both large amounts ofCAPE and strong low-level vertical wind shear. Theseconditions are necessary in that the development of anelevated rear-inflow requires the production of astrongly buoyant plume of air within the updraft cur-rent spreading rearward above the cold pool. The CAPEis important in this regard in that it ultimately controlsthe maximum potential temperature excess that canbe realized within a parcel of air rising within the up-draft. The vertical wind shear is additionally importantin that potential temperature excesses within the up-draft current are maximized when near-surface air islifted quickly to its level of maximum buoyancy,thereby reducing the mixing along the parcel's path.As discussed previously, the strongest lifting of near-surface air is produced when there is strong low-levelvertical wind shear to help balance the cold pool-gen-erated circulation. If the contributions to the generationof the rear-inflow jet due to the warm plume of air aloftare similar in magnitude to the contributions due tothe cold pool near the surface, then the rear-inflow jetmay remain elevated to near the leading edge of thesystem. However, if the contributions from the coldpool are significantly larger than those from the warmplume aloft, then the rear-inflow jet descends andspreads along the surface well behind the leading edgeof the gust front, leading to a shallower, weaker system-scale circulation. These environmental constraints onthe production of long-lived bow echoes are discussedin more detail in section 6.

5. The origin and role of the bookend vortices

The goal of this section is to trace the source of thebookend vortices and to discern what role they mightplay in the initiation and enhancement of the bow echocirculation. This analysis is guided by the three-di-mensional vorticity equation for inviscid Boussinesqmotion:

dwm= w-Vv + V X (Bk), (9)

where a) is the vorticity vector and B represents thebuoyancy field, as defined by (2). The first term onthe right represents the effects of tilting and stretching,which alter an existing vorticity field, while the secondterm represents the generation of horizontal compo-nents of vorticity by horizontal gradients in the buoy-ancy field. In describing the evolution of the bookendvortices, we will be especially interested in the vertical


component of the vorticity equation, which is givenby

dt dz(10)

where toH represents the horizontal components of thevorticity vector, f = dv/dx — du/dy is the vertical com-ponent of vorticity, and the first and second terms onthe right represent the usual tilting and stretching ef-fects, respectively.

a. The origin of the bookend vortices

In a horizontally homogeneous nonrotating fluid,vertical vorticity is generated by tilting the horizontalvorticity a/H inherent in the ambient flow. For example,an isolated updraft growing in a vertically sheared en-vironment will deform the horizontal vortex lines in-herent in the ambient shear upward, producing cy-clonic and anticyclonic vortices on the flanks of thecell (Fig. 15a). Once vertical vorticity is created, thenthe stretching terms may act to either increase or de-crease the existing vorticity. The subsequent develop-ment of a downdraft will deform this pattern consid-erably as the downdraft gradients provide an additionalsource of tilting and the downdraft accelerations stretchthe vertical vorticity previously produced by the up-draft. In conditions of sufficiently strong environmentalvertical wind shear and CAPE, cell splitting may occur,resulting in two mirror-image storms with the rightmember of the splitting pair consisting of a cyclonicupdraft and anticyclonic downdraft at midlevels andthe left member consisting of an anticyclonic updraftand cyclonic downdraft (Fig. 15b). This process is rec-ognized as fundamental to the development of supercellstorms (e.g., Klemp 1987).

The initial evolution of the vertical vorticity field forthe idealized bow echo corresponds closely to this sim-ple model. Between 60 and 120 min, the initial cellsplits into right- and left-moving cells, with the right-moving cell exhibiting a cyclonic updraft and anticy-clonic downdraft, etc. Figure 16 displays the system-relative flow and 8e at 2.5 km AGL at 90 min, whenthe split cell has become well established. A series ofvortex lines emanating from the bookend vortex, aslocated in Fig. 16a, is presented in Fig. 16b. At thistime, most of the vortex lines originate in the ambientshear layer to the south of the storm, and are tiltedupward and downward as they encounter the updraft-downdraft couplet. The location of the anticyclonicvortex on the gradient between high and low 8e is con-sistent with the conservation of equivalent potentialvorticity, as discussed by Rotunno and KJemp (1985).Since the 8e surfaces and ambient vortex lines are bothoriented horizontally in the initial state, the ambientvortex lines will continue to lie within the original 8esurfaces as the flow evolves (assuming no mixing), andthey will both be deformed similarly by the updrafts

and downdrafts. This results in the ambient vortex linestilting vertically through zones of horizontal gradientsof 9e, producing the observed vertical vorticity pattern.

As the system expands, the original split updraft isreplaced by a quasi-two-dimensional line of updraftsextending along the leading edge of the gust front. Asa result, the well-defined cyclonic eddy coincident withthe updraft at 90 min is not evident at later times,although regions of weaker cyclonic vorticity are stillproduced within the new, more short-lived updraftsgenerated along the spreading outflow. However, thebookend vortex that originates in the splitting down-draft is prominent throughout this process. Figure 17presents this structure at 180 min, when the systemhas evolved into the quasi-steady bow echo. Althoughthe bookend vortex is again located within the hori-zontal gradients of 6e, only the warm side of the vortexis now associated with vortex lines that originated inthe ambient shear layer. The remaining vortex linescircle back toward the core of the bow echo, producinga three-dimensional vortex ring centered on the ele-vated rear-inflow jet.

Figure 18 more clearly depicts the association be-tween the vertical vorticity field at 2.5 km AGL andthe vertical velocity field, along with calculations of thestandard tilting and stretching terms as defined in (10).At 90 min, the region of positive vertical vorticity iswell correlated with the region of strongest updraft,while the region of negative vertical vorticity is locatedon the gradient between the updraft and downdraft,although mostly within the downdraft (Fig. 18a). Thetilting term is the primary contributor to the positivevorticity within the updraft at this height (Fig. 18b),with weaker contributions from the stretching terms.For the downdraft region, the tilting contributions tothe negative vertical vorticity are much weaker, withthe primary contributions now comming from thestretching terms. An analysis in which the vorticityforcing terms are integrated along parcel trajectories(not shown) confirms that the relative magnitude ofthe forcing depicted in the horizontal cross sections isalso representative of the relative integrated forcingcontributions.

By 180 min, the zone of negative vorticity associatedwith the bookend vortex has increased substantially insize, although the maximum magnitude of the vorticityhas decreased (Fig. 18d). The positive vorticity asso-ciated with the updrafts is also much weaker, with thestretching and tilting terms also displaying similar butweaker contributions to the vorticity field than earlier(Fig. I8e, f). Small zones of more intense vorticity andvorticity forcing are evident behind the leading edgeof the updraft, closer to the symmetry axis, but thesefeatures are transitory and are not considered further.

The source of the vertical vorticity for the bookendvortex can be clarified further through a direct appli-cation of the circulation theorem; for example, Ro-tunno and KJemp (1985):

15 FEBRUARY 1993 W E I S M A N 659

(a)

(b)

FIG. 15. Schematic depicting how a typical vortex tube contained within (westerly) environ-mental shear is deformed as it interacts with a convective cell (viewed from the southeast).Cylindrical arrows show the direction of cloud-relative airflow, and heavy solid lines representvortex lines with the sense of rotation indicated by circular arrows. Shaded arrows represent theforcing influences that promote new updraft and downdraft growth. Vertical dashed lines denoteregions of precipitation, (a) Initial stage: vortex tube loops into the vertical as it is swept into theupdraft. (b) Splitting stage: downdraft forming between the splitting updraft cell tilts vortex tubesdownward, producing two vortex pairs. The barbed line at the surface marks the boundary of thecold air spreading out beneath the storm (from KJemp 1987).

— = Bk-dl, (11)at Jc

where the circulation C is defined as

C= \ y-d\= f (VX y)-dA (12)Jc Js

and the integration is taken about any closed circuit Con a given surface S. Following the approach of Ro-tunno and Klemp (1985), a series of closed circuitshave been specified that encompass the bookend vor-tices for a sequence of times between 120 and 240 min.Each circuit is traced backward in time by calculatingtime-dependent parcel trajectories at 1-km intervals


T = 9 0 m i n

(a) Vectors, 9E Z = 2.5 km

35

X (km)

(b) Vortex Lines

FIG. 16. Bookend vortex structure at t = 90 min for the idealizedbow echo, (a) Horizontal cross section of flow vectors and 8eat 2.5km AGL. Vectors are presented at every other grid point, with avector length of two grid points representing a magnitude of 25 m s"1.A domain speed of u = 22.5 m s"' has been subtracted from the flowfield. The ground-relative system propagation speed at this time is22 m s~'. 6e greater than 332 K is darkly shaded, while 9e less than326 K is lightly shaded, (b) Vortex lines traced from the locationswithin the bookend vortex denoted by the dots in (a). The sense ofthe vorticity vector is indicated by the arrows. The thick dashed linedenotes the location of the surface gust front.

around the circuit using data saved every 5 min. Thecirculation and the circulation forcing, as denned by(11) and (12), is then calculated at 100-sec intervalsfor each circuit through the analysis period. In the ab-sence of horizontal vorticity generation and mixing ef-fects, the circulation about the circuit will be conserved,and the source of the vertical vorticity must be tracedultimately to the tilting of horizontal vorticity inherentin the ambient shear. If horizontal vorticity is generatedalong a vertical portion of the circuit over the analysisperiod, then the circulation will not be conserved andshould change at the rate specified by the circulationforcing term. In general, the circuits remained well de-fined for about 20 min prior to each analysis time, asmixing effects (both physical and numerical) began todominate beyond this period. However, 20 min wassufficient to clearly identify the source region for the

air parcels making up each circuit and thereby offerssome insights into the source for the circulation as well.

The range of results is well represented by the anal-ysis of the bookend vortex circuit that is specified at180 min, as identified by the rectangle in Fig. 17. Thehistory for this circuit is presented in Fig. 19. The airparcels that are incorporated into the circuit originateprimarily from two sources. One set of air parcels (lo-cated between B and C on the figure) originates nearthe surface, ahead of the system, and is characterizedby high values of 8e. Another set of air parcels (locatedbetween A and D on the figure) originates in the 3-5-km layer to the rear of the system and descends inconjunction with the rear-inflow jet. These air parcelsare characterized by low values of de.

A time series of the circulation along with the in-tegrated contributions from the circulation forcing termfor this circuit are presented in Fig. 20. This analysissuggests that the circulation increased in strength by30% during the prior 20 min (solid line), with the in-tegrated forcing (dashed line) contributing much ofthe observed variation. This would suggest that hori-zontal vorticity generated within the convective systemmay have contributed significantly to the strength of

-60

X (km)

(b) Vortex Lines

FIG. 17. Same as Fig. 16 but at t= 180 min. The ground-relativesystem propagation speed at this time is 27 m s"1. The box in (a)denotes the location of a circuit that is followed over time in Fig. 19.


the bookend vortices at 180 min. For instance, suchhorizontal vorticity generation is largely responsible forthe development of the strong rear-inflow jet, as dis-cussed in section 4. The question arises, however, asto how such horizontal vorticity could be tilted sys-tematically to produce the bookend vortex feature.

One scenario in which convectively generated hor-izontal vorticity might contribute systematically to theproduction of the bookend vortex is if air parcels flow-ing into the vortex traveled along a horizontal buoyancygradient while they ascended or descended into thevortex. Thus, horizontal vorticity that is generatedalong the parcels' path could be systematically tiltedto contribute to the vortex. This concept of the gen-eration of streamwise vorticity has been used, for ex-ample, to explain the preferential development of low-level rotation within supercell updrafts (e.g., RotunnoandKlemp 1985).

In the present scenario, the only parcel paths thatcould be identified that could contribute in the propersense to the development of the bookend vortex werethose parcels that started near the surface ahead of thesystem and that traveled along and then over thesouthern, curved portion of the cold pool before de-scending into the vortex (e.g., parcels between pointsB and C in Fig. 19). Further analysis, however, failedto confirm this as a significant vorticity source for thevortex. Thus, a more careful analysis may be neededto clarify the role of the convectively generated hori-zontal vorticity in producing the bookend vortexstructure. AH in all, the most significant signal fromthis analysis is the relative conservation of the circu-lation for the circuit over the 20-min period, suggestingthat tilting and subsequent stretching of the horizontalvorticity inherent in the ambient vertical wind shearis the primary contributor to the production of thebookend vortices. This hypothesis is supported furtherin section 6, where we note that significant bookendvortices are produced only in simulations in which theambient vertical wind shear is of at least moderatestrength (thereby supplying a continual, significantsource of horizontal vorticity to be tilted and stretched).

b. The role of the bookend vortices

Since the source for the bookend vortex circulationis largely distinct from that producing the two-dimen-sional vertical circulation described in section 4, onemight ask what the relative role of these vortices is inproducing the steady-state bow echo structure. For thispurpose, it is useful to draw an analogy between thebookend vortices and idealized, two-dimensional vor-tex couplets. A schematic representing the basic prop-erties of a vortex couplet is presented in Fig. 21. Thevelocity field of the couplet may be interpreted as thevector sum of the velocity field induced by each vortexindividually. The net flow field includes a jet of airbetween the two vortices (where the two flow fields

add) and much weaker flow outside the vortices (wherethe flow fields subtract). The enhancement of the flowbetween the vortices can be interpreted as a focusingeffect of the vortex couplet. The pressure field is char-acterized by pressure deficits coincident with the vor-tices, which are dynamically balanced with the rotatingflow, and a zone of slightly lowered pressure extendingbetween the vortices, which produces a pressure gra-dient that is consistent with the acceleration of the flowbetween the vortices. This configuration is qualitativelysimilar to the flow and dynamic pressure pattern pre-sented at 180 min for the idealized bow echo (e.g.,Figs. 12 and 18d).

A vortex couplet propagates as the flow field inducedby each vortex advects the other along. The motion ofthe vortex couplet and the strength of the jet dependson the prescribed sizes and magnitudes of the vorticesas well as the spacing between them. By comparing theflow field of vortex couplets of various sizes and mag-nitudes to the flow field realized in the full bow-echosimulation, we can estimate to what extent the bookendvortices contribute to the overall circulation.

For this purpose, we employ a simple time-depen-dent two-dimensional vortex model, as described inthe Appendix. The bookend vortices depicted in thesimulation are approximated by placing two circularregions of vorticity having opposite signs at variousdistances apart. The radius of each vortex is set at 7km, roughly approximating the mean size of the vor-tices between 120 and 240 min in the idealized sim-ulation, with the vorticity within each vortex increasingsmoothly to a maximum in the center. Note that thesize of the vortex is defined by the size of the region ofsignificant vertical vorticity rather than the size of theapparent region of circulation, which may be muchlarger (e.g., Fig. 18d). The spacing between the vorticesis varied between 20 and 60 km, with the maximummagnitude of the vorticity ranging up to 0.015 s~'.The resolution of the vortex model is set at 2 km, tocompare directly with the full simulation.

Figure 22 depicts the magnitude of the jet inducedon the axis between the idealized vortices for the spec-ified range of conditions. Also presented is the ap-proximate magnitude and spacing of the bookend vor-tices along with the observed rear-inflow strength onthe symmetry axis at 2.5 km AGL, presented every 30min between 150 and 240 min, for the full simulation.These results suggest that an idealized, two-dimensionalvortex couplet with the characteristics of the bookendvortices would induce between 7 and 12 m s " 1 of rearinflow on the symmetry axis. This represents 40%-50%of the observed strength of the rear inflow at the analysistimes. Since the analogy between the bookend vorticesand an idealized vortex couplet is at best qualitative,these results must be viewed with some caution. How-ever, trajectory calculations as presented in section 3bsuggest that nearly 30% of the acceleration of the rear-inflow jet near the symmetry axis of the simulation is

662 JOURNAL OF THE ATMOSPHERIC SCIENCES

T = 90min T = l80min

VORTICITY, W

VOL. 50, NO. 4

- 50

(O

- 5 048

STRETCHING, VORTICITY

= 2.5 km - -

98 48

( f )

= 2.5 km -

98X (km) X(km)

15 FEBRUARY 1993

10

Z(km)

Y(km)

-60

WEISMAN

(a) 160 Mm

663

X(km)

FlG. 19. Evolution of the bookend vortex circuit depicted in Fig.17, ending aXt = 180 min. The circuit is traced at (a) 160 min, (b)170 min, and (c) 180 min, with the corner points of the circuit,labeled A, B, C, and D, marked at each time for reference. The senseof the circuit is positive in the counterclockwise direction. The thickdashed line denotes the location of the surface gust front.

associated with dynamically induced horizontal pres-sure gradients, with these contributions increasing tonearly 50% along the flanks of the system. A significantportion of this dynamical pressure gradient appears tobe associated with the bookend vortices. This suggeststhat the bookend vortices do play a significant role inenhancing the strength of the bow-echo circulation.

A similar analysis of the induced propagation speedsfor these idealized vortex rings (not shown) suggestmagnitudes of only 2ms""1 . The propagation speedsfor the full bow echo, however, range between 25 and30 ms" 1 . Thus, the propagation that might be induced

- 1 . 2 - •

160 170

TIME (min)

180

FIG. 20. Time series of circulation (solid) and integrated circulationforcing (dashed) for the circuit described in Fig. 19, which is followedfrom 160 through 180 min.

by the bookend vortices does not appear to contributesignificantly to the system motion.

6. Environmental conditions for bow-echo genesis

The hypotheses put forth in the preceding sectionsto explain the development of the idealized bow echodepend on both the strength of the low-level verticalwind shear and the amount of thermodynamic insta-bility in the environment ahead of the system. Thevertical wind shear controls the characteristics of thelifting that is realized at the leading edge of the gustfront as well as the ability to generate the bookendvortices at the edges of the system. The thermodynamicinstability, however, controls ultimately the maximumamount of buoyancy that can be realized within theupdraft current that spreads rearward above the surfacecold pool, which is a crucial factor in the developmentof a strong, elevated rear-inflow jet. This suggests thatconvective systems with characteristics similar to theidealized bow echo can exist for a rather wide range ofambient thermodynamic and vertical wind-shear con-ditions, as long as certain minimum thresholds for eachof these environmental parameters is achieved. Thegoal of this section is to more clearly establish whatthese thresholds are for the idealized simulations andto compare these results to the environmental condi-tions associated with observed severe, long-lived bowechoes.

FlG. 18. Horizontal cross sections at 2.5 km AGL depicting vertical vorticity, tilting, and stretching, along with vertical velocity and tlowvectors at [(a), (b), and (c)] 90 min and [(d), (e), and (f)] 180 min. For (a) and (d), vertical vorticity is contoured every 20 X 10~4 s~',with regions of vertical velocity greater than 4 m s"' darkly stippled and regions of vertical velocity less than - 2 m s"1 lightly stippled.Vectors are presented every other grid point, with a vector length of two grid points representing a magnitude of 25 m s~'. A domain speedof 22.5 m s"' has been subtracted from the flow field. For (b), (c), (e), and (f), the tilting and stretching terms are contoured every 8X 10"6 s~2, respectively, with regions of vertical vorticity greater than 20 X 10~4 s~' darkly shaded and vertical vorticity less than - 2 0

664 J 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 VOL. 50, No. 4

N— N

\/

/

^z 7 s\%

FlG. 21. Schematic representation of an idealized two-dimensionalvortex couplet, depicting the stronger flow induced between the vor-tices. Shading denotes regions of constant vorticity. Unshaded regionshave zero vorticity. Dashed contours represent an approximate neg-ative pressure perturbation field that is consistent with the flow pattern.The L denote the locations of the lowest pressure.

For this purpose, the ambient wind and thermody-namic profiles are varied over a wide range of condi-tions associated with severe convective systems. Thethermodynamic profile is varied by changing both thelapse rate through the troposphere, as depicted in Fig.

INDUCED STRENGTH OF VORTEX JET

20 40

VORTEX SPACING (km)

FIG. 22. Induced strength of the jet between the vortices for anidealized two-dimensional vortex couplet, given the spacing betweenthe vortices and the strength of the vortices. The observed strengthof the bookend vortices at 150, 180,210, and 240 min for the idealizedbow echo are included in the figure, along with the observed strengthof the rear-inflow jet between the vortices at those times (numbersin parentheses, meters per second, taken relative to the ambient 2.5-km flow).

3, and the low-level mixing ratio. This produces vari-ations in the CAPE, level of free convection (LFC),and the amount of negative area to be overcome by arising parcel. The range of conditions covered by theseexperiments is presented in Table 1, and includes therange of observed conditions associated with bow-echoevents. Two different wind-shear profiles are also con-sidered. The first restricts the shear layer to the lowest2.5 km AGL with constant winds above, as for thesimulations discussed previously. The second extendsthe shear layer to 5 km AGL. Each shear profile isvaried in magnitude from Us = 5 m s~' to Us = 40m s~' for the maximum wind over the prescribeddepth. As before, a single cell is initialized on the sym-metry axis of the model domain and is allowed toevolve through 240 min of simulation time.

a. Simulation results

Three basic types of convective organization areidentified in these experiments, as depicted schemati-cally in Fig. 23. The first category, designated "W,"consists of cases in which the cold pool circulationclearly overwhelms the ambient vertical shear. Thisleads to an upshear-tilted system of rain cells tens ofkilometers behind the leading edge of the gust front.These systems characteristically develop strong rear-ward flow above the cold pool, with only weaker, dis-organized midlevel rear inflow into isolated cells at theback edge of the system. The rear inflow descendsquickly to the ground to produce a wide zone of system-relative surface outflow extending to the leading edgeof the gust front. Portions of these systems may appearbow shaped for short periods of time and may alsoproduce periods of strong surface winds (especiallyearly in the evolution, in association with strong, iso-lated convective cells), but these features tend to dis-sipate quickly, and the structures described for theidealized long-lived bow echo (e.g., an elevated rear-inflow jet and bookend vortices) do not develop. Anexample of this evolution is the moderate-shear casedescribed by Weisman (1992).

A second category, designated "B," represents casesthat develop a long-lived bow echo structure, similar

TABLE 1. Thermodynamic conditions used forenvironmental sensitivity experiments.

CASE

1234567

Low-levelmixing

ratio(g kg"')

14131415131415

Cloud

base(m)

110712521107963

12521107963

Level offree

convection(m)

1907195217071363175215071263

CAPE

Negative

- 1 7- 1 6- 1 2

- 5- 1 0-6- 2

(m2 s-2)

Positive

1182179523773011318738474549


a) Weak Cells

("W")

b) Bow Echo

("B")

c) Split Cells

("S")

FlG. 23. Schematic depiction of the primary modes of convectiveorganization identified in the sensitivity experiments between 180and 240 min. Thick and thin arrows represent system-relative flowat 2.5 km AGL and the surface, respectively. Contours representrainwater concentration. Thick barbed line represents the locationof the surface gust front.

to that described for the idealized experiment. Suchcases are characterized by a 40-80-km bow-shaped arcof rain cells along the leading edge of the system. Astrong, elevated rear-inflow jet extends to the leadingedge of the gust front, with strong rearward flow at theends of the bow forming bookend vortices. Strong sys-tem-relative surface outflow is confined to a narrow(5-10-km) zone just behind the gust front. An inter-mediate category, designated "I ," represents cases thathave attributes of both cases W and B (not includedin Fig. 23).

A third category, designated "S," is characterized bystrong, more isolated, long-lived cells scattered alongthe leading edge of the gust front. Many of these cellsare supercellular, consisting of rotating updrafts andhaving evolved through a splitting process. Each cellmay induce strong, midlevel jet features associated withcell-scale processes, but a larger-scale organized rearinflow, as occurs with the idealized bow echo, is notevident.

The form of organization that dominates these sim-ulations between 180 and 240 min is depicted in Fig.24. The cases with Us= 15 m s~' or less over the pre-scribed depth all evolve as described for the disorga-nized systems, independent of the amount of instabil-ity. As the shear increases, more organized structuresoccur, with bow echoes dominating the stronger-insta-bility part of the range and split supercells favored forthe weaker-instability cases or the strongest deep-shearcases. Recall, however, that the cases that eventuallyevolve into bow echoes all originate with splitting cellsearlier in the lifetime of the system, as described forthe idealized bow echo discussed earlier. It is possiblethat some of the supercell cases at 240 min could alsoevolve into bow echo structures at later times if thecold pools continue to strengthen and deepen. As thedepth of the shear layer increases, the range of condi-tions that produce bow echoes shrinks to the higher-instability, intermediate-shear range of the spectrum.Tn general, well-defined, long-lived bow echoes occur

HIa.<<o

4000

3000

2000

1000

4000

3000

2000

1000

W

M

W

W

-

SYSTEM ORGANIZATION:

M

M

W

W

W

W

W

W

W

-

(a) 2.5 km Shear

H

N

K

-

(b) 5 km Shear

B

B

B

1

I

S

1

I

I

1

B

B

B

B

S

s

180-240 min

B

B

B

S

S

B

B

S

S

-

-

-

-

s

s

s

-

-

10 20

Us(ms-i)

30 40

FIG. 24. Primary modes of convective organization identified forthe (a) 2.5-km shear and (b) 5-km shear experiments between 180and 240 min for each simulation. Us represents the maximum mag-nitude of the wind for each wind profile. In the figure W representsa weak upshear-tilted system, I represents an intermediate form oforganization, B represents a bow echo, and S represents split supercells,as described in Fig. 23 and in the text; "—" indicates that the stormsystem had dissipated by 180 min.


for shear magnitudes of at least 2 0 m s ' over the pre-scribed depth and CAPEs of at least 2200 m2 s"1. Inretrospect, the idealized bow echo case represents aboutthe minimum CAPE that produces this type of con-vective structure within these simulations.

One of the more pertinent characteristics of bowechoes and, indeed, the one that created much of theobservational interest is their propensity to producelong swaths of damaging surface winds. Figure 25 pre-sents the maximum ground-relative wind realized atthe lowest level of the model (350 m) at 240 min foreach simulation. The strongest winds at 240 min gen-erally occur for both stronger shears and strongerCAPE. If we consider maximum 350-m winds of 35m s~' or greater as indicating significant potential forproducing damaging surface winds, then we note thatsuch systems occur for all cases of CAPE greater than2000 m2 s~2 and shears greater than 15ms" 1 , includ-ing but not restricted to the cases that have evolvedinto structures similar to the idealized bow echo. Sincea free-slip condition has been specified at the lowerboundary, these model-produced near-surface windsmay be a slight overestimate of the actual surface windspeeds that would be observed in such systems. HOW-

M A X I M U M NEAR-SURFACE WIND (ms1)T = 240 min

4000 -

3000 -

2000 -

1000 '

LJJ

a.

4000

3000 -

2000 -

1000

1— 1

28 35

26 31

22 30

0 0

i

40

41

40

35

27

0

(a) 2.5 km Shear

37

30

10

0

(b)5 km Shear

i

12

45

11

35

31

11

i

11

39

36

30

15

40

11

39

31

16

43

42

36

36

27

i

39

35

36

22

••• v • - ' • " • ' •

45

24

22

0

10 20 30U,(ms-i)

40

FIG. 25. Maximum near-surface (350 m AGL) wind (m s ') forthe (a) 2.5-km shear and (b) 5-km shear experiments at t = 240min. U, represents the maximum magnitude of the wind for eachwind profile.

ever, the inclusion of surface drag in such simulations(e.g., Wilhelmson and Chen 1982) does not signifi-cantly effect the overall system evolution, suggestingthat the general dependence of near-surface wind mag-nitudes on environmental vertical wind shear andCAPE would also remain intact. Future studies,though, must more carefully consider these effects.

b. Comparison with observations

A qualitative comparison between these model-de-rived criteria and the results from the observationalcases discussed in the Introduction suggests many sim-ilarities. For instance, both the present study and thederecho climatology clearly indicate that large amountsof instability and moderate-to-strong magnitudes ofenvironmental vertical wind shear are necessary forsuch severe, long-lived wind events. However, theminimum magnitude of the wind shear for the observedderechos is slightly less than that for the simulated bowechoes; that is, at least 15 m s~' of speed variationfrom the surface to 700 mb was observed for the der-echos, while a 20 m s"' threshold was suggested forthe long-lived, idealized bow echoes. This apparentdiscrepancy, though, may merely reflect the observa-tional tendency to identify derecho events based onsurface winds alone, rather than on the long-livedstructural characteristics emphasized in the presentstudy. As discussed before, if the present simulationsare categorized based on surface winds alone, then theminimal vertical shear conditions for the productionof long-lived convective systems with strong surfacewinds does decrease to the 15 m s " 1 of low-level windvariation noted for the derechos. This suggests that avariety of mesoconvective structures can be associatedwith long-lived, damaging surface wind events, withthe idealized bow echo described herein representingan extreme example that occurs only for the stronger-shear cases.

As a more specific example from the Introduction,the 28 May case described by Burgess and Smull (1990)exhibits 4200 m2 s~2 of CAPE with about 17 m s"1 ofshear over the lowest 2 km AGL, extending in a morecomplicated fashion over greater depths. These envi-ronmental conditions fall well within the range of con-ditions that produce severe near-surface winds withinthe simulations, but the vertical shear magnitudes aremore characteristic of the intermediate category ofconvective structures rather than of the long-lived bowechoes. Consistently, this particular convective systemevolves much more quickly than the idealized bowecho, with a total lifetime of approximately 3 h.

The environmental conditionsfor the bow echoesdescribed by Schmidt and Cotton (1988), Johns andLeftwich (1988), and Smith (1990) again exhibit ther-modynamic instabilities well within the range of modelconditions conducive to the development of long-livedconvective systems with strong surface winds (CAPEs


between 2500 and 4000 m2 s 2). However, these casesalso exhibit between 20 and 30 m s " 1 of vertical shearbetween the surface and 5 km AGL, which also clearlyfalls into the range of model conditions that producedlong-lived bow echoes. Consistent with the model re-sults, these convective systems remained coherent forbetween 6 and 12 h and, in addition, also exhibitedsupercell-type structures early in their lifetimes. Thus,there seems to be much similarity in the range of struc-tures and associated environments for convective sys-tems that produce strong surface winds over extendedperiods of time for both the idealized simulations andthe observations.

7. Summary and discussion

We have documented herein the tendency for sim-ulated convective systems triggered in environmentsof large CAPE and strong low-level vertical wind shearto evolve into bow-shaped systems of cells, similar toobservational descriptions of long-lived bow echoes.For some cases, this evolution merely reflects the en-hanced ability of a cold pool spreading in a verticallysheared environment to regenerate new convectivecells, as described by RKW, WKR, and Weisman andKJemp (1986). In such cases, the bow shape and lon-gevity of the convective system is directly attributed tothe preferential and continual regeneration of convec-tive cells along the downshear portion of the spreadingcold air. In certain stronger-shear simulations, however,a distinct, quasi-steady structure evolves 3-4 h into thelifetime of the convective system that has yet to bedescribed in numerical studies. This structure consistsof a 40-100-km long bow-shaped system of cells withcyclonic and anticyclonic eddies (bookend vortices) at2-3 km AGL at the northern and southern ends of thesystem, respectively, and a strong rear-inflow jet be-tween the vortices. This rear-inflow jet remains elevatedto near the leading edge of the convective system andplays a major role in enhancing the lifting of the po-tentially warm surface air ahead of the system, thusmaintaining the convective circulation. The similaritiesbetween this numerically generated feature and theavailable observations of many severe long-lived bowechoes suggest that such systems may indeed representa dynamically unique form of mesoconvective orga-nization. It is this new, coherent structure that has beenthe focus of the current investigations.

The evolution to this new quasi-steady structure oc-curs systematically as the cold pool strengthens anddeepens in response to the retriggering of convectivecells along the leading edge of the system. As describedby RKW, the cold pool-generated circulation graduallyovercomes the circulation inherent in the ambient ver-tical wind shear, and the system begins to tilt upshearand weaken. For conditions of large CAPE and stronglow-level vertical wind shear, however, the upsheartilting of the system results in the production of strong

horizontal gradients of positive buoyancy aloft nearthe back edge of the system that generate a strong, el-evated rear-inflow jet. This elevated rear inflow helpsto reestablish a balance between the cold pool circu-lation and the ambient vertical wind shear, promotingdeep forced lifting at the leading edge of the system upto jet level. Above the jet, the updraft current is thenforced rearward above the spreading cold pool, therebymaintaining the buoyant plume of air aloft necessaryfor the maintenance of the rear-inflow structure.

The generation of this intense rear-inflow jet is alsoaided by the development of bookend vortices, whichhelp to initiate and focus the rear inflow into the coreof the developing system. Indeed, a long-lived bowecho-type system does not develop in these simulationswithout the prior existence of these features. Throughan analogy with idealized two-dimensional vortexcouplets, it was determined that these bookend vorticesmay contribute between 30% and 50% of the resultantrear-inflow strength during the system's mature phase.The primary source for this circulation is the tiltingand subsequent stretching of horizontal vorticity in-herent in the ambient shear layer ahead of the system,as horizontal vortex lines are deformed upward andthen downward by the updraft-downdraft pattern nearthe ends of the convective line segment.

For the idealized bow echo, the bookend vorticesoriginate through the splitting of the initial isolated celland are maintained as the system expands to form aquasi-two-dimensional convective line. For the squallline example depicted in Fig. 2, however, the bookendvortices develop in situ at the ends of a preexistingconvective line segment. This suggests that cell splitting,per se, is not crucial for the development of these fea-tures. In particular, bookend vortices may evolvemerely as a consequence of there being physical endsto the convective system; that is, the updraft-downdraftcouplet at these locations will always deform the am-bient vortex lines so as to produce this flow pattern.This suggests that the development of these circulationfeatures in squall line-type scenarios may dependsomewhat randomly on whether the convective systemjust happens to evolve into separate convective linesegments of sufficient strength to promote the furtherdevelopment.

The length of the resultant convective segment mayalso be important in this regard. As discussed in section5b, the magnitude of the "focusing" effect for thebookend vortices decreases with increasing distancebetween the vortices. Thus, vortices that are spacedtoo far apart may not generate sufficient rear inflow toinitiate the bow echo evolution. On the other hand,the smaller bow echo structure to the north in Fig. 2bwas not maintained nearly as long as the larger, mainbow echo feature. This suggests that an optimal lengthscale may exist for the development of the longest-lived bow echoes. This possibility will be investigatedin future studies.


An extensive set of environmental sensitivity exper-iments has been completed to more clearly documentthe range of environmental conditions that would sup-port bow echo genesis. These simulations suggest thatconvective systems similar to the idealized bow echomay be generated in environments in which the CAPEis at least 2000 m2 s~2 and the vertical wind shear isat least 20 m s " ' over the lowest 5 km AGL. In addi-tion, the development of such systems is especially fa-vored if most of this vertical wind shear is confined tothe lowest 2.5 km AGL. The emphasis on low-levelvertical wind shear occurs since such conditions pro-duce the optimal conditions for lifting at the leadingedge of the gust front. Strong vertical wind shear alsosupplies a significant source of horizontal vorticity tobe tilted and stretched at the ends of the convectivesystem for the production of the bookend vortices.These environmental conditions correspond closely tothose that have been associated with observed severe,long-lived bow echoes and derechos.

The emphasis in the observational studies of bowechoes has been on the association between these fea-tures and the production of long-lived swaths of dam-aging surface winds. The current simulations reconfirmthis association, but also suggest that long swaths ofdamaging surface winds may develop for a wider rangeof environmental conditions than those associated withthe idealized bow echo. For the present simulations,near-surface winds of 30 m s~' or greater were stillbeing produced 4 h into the system's life for environ-mental CAPEs of at least 2000 m2 s~2 and wind vari-ations of at least 15 m s"1 over the lowest 2.5-5 kmAGL (20 m s~' of shear was necessary for the idealizedbow echoes). Since these weaker shears are more com-monly observed in association with such large amountsof CAPE, many of the observed windstorm events mayfall within this intermediate range of vertical wind-shearvalues. The convective systems in such cases, however,evolve more quickly than the stronger-shear cases, asthe rear-inflow jet descends to the surface well behindthe leading edge of the system, thereby promotingshallower lifting along the gust front.

The convective structures that are produced in thesesimulations are purely symmetrical along an axis par-allel to the vertical wind-shear vector. This result ispredetermined by using a unidirectional vertical wind-shear profile for the initial state and by not includingthe Coriolis force. As depicted in Fig. 1, however, ob-served bow echoes may evolve from a symmetricalstructure early in their life cycle to one in which thecyclonic bookend vortex at the northern end of thesystem dominates over the anticyclonic bookend vortexat the southern end. This eventually produces acomma-shaped appearance to the convective rainfallpattern. Preliminary simulations suggest that the in-clusion of the Coriolis force does result in the produc-tion of a dominant cyclonic vortex at the northern endof the system by 6 h, similar to the observational model.

However, the basic dynamical structures describedherein for first 4 h of evolution of the idealized bowecho still remain intact. These results, along with theeffects of including directionally varying vertical windshears, will also be discussed in a future study.

Another limitation to the present results is the lackof an ice phase in the model formulation. For instance,Fovell and Ogura (1988) noted that the inclusion ofice in two-dimensional squall line simulations en-hanced the rearward flux of precipitation particles intothe trailing portions of the storm, producing a moreextensive and realistic stratiform precipitation area. Icealso tended to increase the positive potential temper-ature perturbations aloft due to the latent heat of fusion,thus helping to generate stronger rear-inflow circula-tions. These effects could have significant impacts onlonger-term simulations (i.e., greater than 6 h), buthave not tended to significantly impact structures pro-duced in shorter-term simulations, as presented here.

Future studies must also consider a wider range ofthermodynamic conditions than presented here, in-cluding variations to the midlevel moisture and low-level stability characteristics. The amount of midlevelmoisture can affect both the strength of the convectiveupdrafts—through the entrainment of drier, more sta-ble air—and the strength of the convective downdraftsand rear-inflow jet—through changes to the amountof evaporation produced at the rear of the system. Thenet affect of these variations on a convective system,however, is difficult to gauge. As noted in the Intro-duction, though, severe, long-lived bow echoes are ob-served in environments characterized by both moistand dry midlevel conditions, suggesting that midlevelmoisture content is not a critical parameter for bowecho development. Further study of the effects of low-level stability on such systems is motivated by the ob-servation that some of these systems are generated andmaintained despite very stable near-surface conditions(e.g., Schmidt and Cotton 1989).

Finally, there is strong need for observations thatcan confirm the structural characteristics produced inthe idealized simulation. The ongoing study of Burgessand Smull (1990) supplies some of the necessary doc-umentation, but additional case studies will be neededto verify the representativeness of these results and toextend them to more strongly sheared environments.The simulation of an observed case for which both theevolutionary and mature phases of the systems are welldocumented would also represent a crucial addition tothis initial study. We hope, however, that the intriguingdynamical characteristics of the idealized bow echoesproduced in these simulations will motivate further in-vestigations of this type of organized mesoconvectivesystem.

Acknowledgments. In completing this work, I havebenefited greatly from conversations with and reviewsof the manuscript by Richard Rotunno, Jim Fankhau-


ser, Dave Parsons, John Clark, J. Michael Fritsch, BillFrank, Hampton N. Shirer, Bill Skamarock, and threeanonymous reviewers.

APPENDIX

A Two-Dimensional Vorticity Model

In order to investigate the basic properties of vortexcouplets, we have devised a simple numerical modelbased on a two-dimensional vorticity equation, whichis derived from the inviscid, nonrotating horizontalmomentum equations and the continuity equation;that is,

du

dt

dv

Tt

duU dx +

dvuFx +

du

dx

duVdy

dvVd~y

dv+ dy

_ - 1

P

_ - 1

P

= 0,

*P + Fdx

dy y

, (Al)

, (A2)

(A3)

where Fx and Fy are momentum source terms that havebeen included to represent some specified flow into orout of the two-dimensional plane. The vorticity equa-tion is derived by cross differentiating (Al) and (A2),taking their difference, and using the incompressibilityassumption to yield

dt dt dt _. .— + u — + v — = F(x, y),dt dx dy

where f is defined as

_ du dv

dy dx

(A4)

(A5)

and F(x, y) is now an assumed distribution of vorticityforcing. Physically, F(x, y) might represent the effectsof localized tilting or advection of vorticity from outsidethe two-dimensional plane.

Since the flow is assumed to be incompressible, astreamfunction may be defined such that

f = V2ip, where u = and v = — , (A6)dy dx

and (A4) may now be rewritten as

dt dy dx (Al)

This equation is solved numerically using a leapfrogtime step and centered-in-space finite differences. ADufort and Frankel time and space smoothing term isalso included at each time step in order to control thegrowth of numerical noise (Haltiner and Williams1980, p. 155).

The model is initialized by specifying an initial fieldof vorticity or vorticity forcing. At each time step, the

streamfunction and thereby the velocity componentsare derived by solving the Poisson equation (A6) re-lating the streamfunction to the vorticity field. ThePoisson equation is solved using a standard successiveoverrelaxation iteration technique (Haberman 1987,p. 516). The streamfunction and vorticity are assumedto be zero on the boundaries of the domain, yieldingmirror-image symmetry across each boundary.

For the investigation of two-dimensional vortexrings, F(x, y) is set to zero, and the vorticity field isinitialized by specifying two circular zones of vorticityof equal magnitude but opposite sign in close proximityto each other. The strength of each vortex is definedby the following formula:

F = Fmix[

RAD, = [(X-XC)2

+ 1 ] ] , (A8)

- Yc)2]1/2, (A9)

where Xc and Yc represent the coordinates of the centerof each vortex. The radius of each vortex (RAD) is setat 7 km, with the magnitude of the vorticity rangingsmoothly from the maximum, Finjt, at the center ofthe circle to 0.15 F\n\t at the edge. Outside the circle,the vorticity is set to zero. The mean flow in the domainis set to zero, and the vortex couplet propagates as eachvortex advects the other along. The domain is set at120 km by 120 km, with a 2-km grid resolution, andthe time step is set at 30 sec. Each simulation is runfor 60 min. These simulations make use of the mirror-image symmetry by placing the right member of thevortex couplet preferentially near the X = 0 boundary.The left member is, thus, only implicitly simulated.Further information concerning these simulations andthe model formulation can be found in Weisman(1990).

REFERENCES

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