‘‘A Theory for Strong Long-Lived Squall Lines’’...

VOL. 61, NO. 4 15 FEBRUARY 2004J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

q 2004 American Meteorological Society 361

‘‘A Theory for Strong Long-Lived Squall Lines’’ Revisited

MORRIS L. WEISMAN AND RICHARD ROTUNNO

National Center for Atmospheric Research,* Boulder, Colorado

(Manuscript received 29 April 2002, in final form 7 August 2003)

ABSTRACT

Based on the analysis of idealized two- and three-dimensional cloud model simulations, Rotunno et al.(hereafter RKW) and Weisman et al. (hereafter WKR) put forth a theory that squall-line strength and longevitywas most sensitive to the strength of the component of low-level (0–3 km AGL) ambient vertical wind shearperpendicular to squall-line orientation. An ‘‘optimal’’ state was proposed by RKW, based on the relative strengthof the circulation associated with the storm-generated cold pool and the circulation associated with the ambientshear, whereby the deepest leading edge lifting and most effective convective retriggering occurred when thesecirculations were in near balance. Since this work, subsequent studies have brought into question the basicvalidity of the proposed optimal state, based on concerns as to the appropriate distribution of shear relative tothe cold pool for optimal lifting, as well as the relevance of such concepts to fully complex squall lines, especiallyconsidering the potential role of deeper-layer shears in promoting system strength and longevity. In the following,the basic interpretations of the RKW theory are reconfirmed and clarified through both the analysis of a simplifiedtwo-dimensional vorticity–streamfunction model that allows for a more direct interpretation of the role of theshear in controlling the circulation around the cold pool, and through an analysis of an extensive set of 3Dsquall-line simulations, run at higher resolution and covering a larger range of environmental shear conditionsthan presented by WKR.

1. Introduction

The association of squall lines with environmentalvertical wind shear has been recognized since routineupper-air soundings first became available (e.g., Byersand Braham 1949; Newton 1950; Bluestein and Jain1985). Indeed, numerical modeling studies (e.g., Hane1973; Thorpe et al. 1982; Rotunno et al. 1988, hereafterRKW; Weisman et al. 1988, hereafter WKR; Fovell andOgura 1989; Robe and Emanuel 2001) show that sim-ulated squall lines depend critically on the ambient ver-tical wind shear. Using a two-dimensional numericalmodel, Thorpe et al. (1982) noted more specifically thatsimulated squall lines were strongest for environmentsin which the vertical shear was confined to low levels.RKW and WKR noted a similar dependence of squall-line strength and longevity on the wind shear, using bothtwo- and three-dimensional simulations, and observedthat the simulated squall line was maintained throughthe regeneration of new cells along a line of cold outflowproduced during the decay of old cells. RKW found thatthe ability of the cold surface outflow of an old cell to

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

Corresponding author address: Morris L. Weisman, NCAR, P.O.Box 3000, Boulder, CO 80307-3000.E-mail: [email protected]

lift environmental boundary layer air to its level of freeconvection (and so produce a new cell) is enhanced withlow-level shear, and concluded that the cold-pool–shearinteraction is a central element in understanding themaintenance of strong squall lines in the absence ofsignificant external forcing features (e.g., cold fronts,etc.). Critics of the RKW theory of squall lines havecharged that the focus on the cold-pool–shear interactionis an oversimplification as it neglects the larger-scalesquall-line circulations (e.g., Lafore and Moncrieff1989; Garner and Thorpe 1992). Others (e.g., Coniglioand Stensrud 2001; Evans and Doswell 2001) contendthat, upon closer inspection, observations and modelsdo not support such a strong connection between low-level shear and squall-line characteristics. Still othershave put forth alternative theories to explain the deeplifting induced by the cold-pool–shear interaction (e.g.,Xu 1992; Xu and Moncrieff 1994; Xu et al. 1996, 1997;Xue 2000). The present paper addresses these issues.

The RKW theory of squall lines is fundamentallyconcerned with how typical thunderstorms, which un-dergo a 30–50-min life cycle (e.g., Byers and Braham1949), can be regenerated along a line. RKW proposeda simple way of thinking about how low-level shearenhances the regeneration process: For a cold poolspreading in an environment without shear (Fig. 1a), thecirculation at the leading edge is as described for aclassic density current (e.g., Simpson 1997), with the

362 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 1. (left) Cold pool spreads away from a decaying convectivecell in an environment with no vertical wind shear. (right) Low-levelvertical wind shear balances cold-pool circulation on the downshearside, enhancing the ability to regenerate convective cells throughdeeper lifting.

FIG. 2. Three stages in the evolution of a convective system. (a)An initial updraft leans downshear in response to the ambient verticalwind shear, which is shown on the right. (b) The circulation generatedby the storm-induced cold pool balances the ambient shear, and thesystem becomes upright. (c) The cold-pool circulation overwhelmsthe ambient shear and the system tilts upshear, producing a rear-inflowjet. The updraft current is denoted by the thick, double-lined flowvector, and the rear-inflow current in (c) is denoted by the thick solidvector. The shading denotes the surface cold pool. The thin, circulararrows depict the most significant sources of horizontal vorticity,which are either associated with the ambient shear or which are gen-erated within the convective system, as described in the text. Regionsof lighter or heavier rainfall are indicated by the more sparsely ordensely packed vertical lines, respectively. The scalloped line denotesthe outline of the cloud. Here, C represents the strength of the coldpool while Du represents the strength of the ambient low-level verticalwind shear, as described in the text (adapted from Weisman 1992).

environmental air being forced up and over a deeperhead region, and then subsiding over the main body ofcold air. When ambient shear is present (Fig. 1b), thecirculation associated with the shear will, on the down-shear side, counteract some of the circulation associatedwith the cold pool, producing deeper lifting there. Thedeepest lifting and largest potential for cell retriggeringoccurs when the cold pool and shear circulations are inbalance. Naturally, the shear layer occupying the samevertical levels as the cold pool was proposed to be themost important for this effect. This basic impact of shearon a density current has been reproduced in many otherrecent studies (e.g., Xu et al. 1996, 1997; Xue 2000),although differing interpretations of the effect have beenoffered, as is discussed below.

The RKW cold-pool–shear interaction theory ad-dresses the threshold question of how the shear affectsthe transformation of ordinary finite-lifetime thunder-storms into a long-lasting system of cells continuallyregenerating along a line. Our view is that only afterthis threshold question has been answered, does it makesense to consider questions concerning the collectiveeffects of many generations of cells. For example, ma-ture squall lines often have a rear-inflow jet (e.g., Smulland Houze 1987) which, according to numerical sim-ulations, can originate through the collective heating/cooling pattern of the convective cells (Lafore and Mon-crieff 1989; Weisman 1992). Consider the evolution ofa squall-line circulation as presented schematically inFig. 2, where C is a velocity representing the strengthof the cold pool and Du represents the magnitude of thelow-level ambient vertical wind shear. Before a signif-icant cold pool develops (C K Du), the convective cellswithin the squall line tilt predominantly in a downsheardirection in response to the ambient shear (Fig. 2a); aftera cold pool has developed, its circulation may counterthat associated with the low-level shear to produce deep-er lifting at low levels and an overall more upright con-vective structure (C ; Du, Fig. 2b); finally, if the coldpool evolves to a state where C . Du, then the circu-lation associated with the cold pool dominates that ofthe shear, thereby sweeping the convective cells andassociated zone of heating/cooling rearward (toward theleft in the figure), where it can induce the generation ofa rear-inflow jet (Fig. 2c). The RKW ‘‘optimal’’ state

is envisioned when C/Du is close to 1, whereby thesystem maintains an upright configuration and the deep-est lifting is produced at the leading edge of the coldpool. However, for all except the most strongly shearedenvironments, squall lines tend to evolve through allthree depicted phases during their lifetimes, as coldpools usually strengthen over time and eventually be-come strong enough to overwhelm the ambient shear.

The induced rear-inflow circulation during the maturephase of a squall line might fundamentally alter the

15 FEBRUARY 2004 363W E I S M A N A N D R O T U N N O

original conditions allowing continual cell regeneration.However, based on the analysis of an extensive set ofquasi-three-dimensional squall-line simulations, WKRand Weisman (1992) showed that the dependence ofsquall-line structure, strength, and longevity on envi-ronmental vertical wind shear could still be largely ex-plained based on the simple cold-pool–shear interactionideas. WKR considered both the ability of a system tomaintain upright convective cells along the leading edgeof the gust front for long periods, and also the totalcondensation produced within the convective system.By both measures, the optimal squall lines occurred forenvironments with moderate-to-strong vertical windshear confined to the lowest 2.5 km AGL and orientedperpendicular to the squall line. Deeper layers of strongshear were able to support the development of strong,isolated cells scattered along the line (some of whichwere supercellular), but were not as effective as theshallower shear layers in producing a continuous lineof updrafts. Weisman (1993) and Weisman and Davis(1998) further showed that strong low-level shears wereespecially conductive to the development of three-di-mensional bow-shaped segments embedded within suchlines, with attendant ‘‘bookend’’ or ‘‘line-end’’ vorticesat midlevels (e.g., 2–6 km AGL) with attendant locallyintense rear-inflow jets that could further enhance sys-tem strength.

Although there is general agreement among the mod-eling studies that low-level shear is important, there isless agreement on why it is important, and moreover,its importance relative to other factors. In diagnosingsimulations of tropical squall lines, Lafore and Mon-crieff (1989) emphasized that, in contrast with RKW, itis the differential speed of the cold pool versus the con-vective cells that is the more critical factor in systemmorphology (reinforcing the interpretation of Thorpe etal. 1982; see also Moncrieff and Liu 1999), and there-fore suggested that ‘‘the wind shear in the entire tro-posphere should be considered in the interaction of thecold pool and the convective cells.’’ They also notedthat system-scale circulation features, such as the abovediscussed rear-inflow jets, contribute significantly to ma-ture squall-line structure, stating further that ‘‘it is anoversimplification to say that the interaction of the shearand the cold pool adequately explains the longevity ofmesoscale convective systems because the organizationof the vorticity field is predominantly on the scale ofthe entire system and not confined to a localized con-vection scale.’’ This issue of ‘‘local’’ versus ‘‘global’’effects is summarized in Lafore and Moncrieff (1990),where they emphasize that many factors, such as theupstream effects of the squall line, which can modifythe low-level shear, the wind profile in mid- to upperlevels, the differential movement of convective cells,and synoptic-scale effects all can have a significant im-pact on system morphology. Thus, they conclude: ‘‘Lo-cal vorticity balance is an interesting concept, but itsimportance to squall line convective systems, relative

to the more complex processes listed above, remainsunresolved.’’

Using a composite environment for observed severe,long-lived convective wind storms (referred to as de-rechos), Coniglio and Stensrud (2001) employed a me-soscale forecast model with mesoscale variability in theenvironmental conditions, and successfully simulatedmany of the essential characteristics of such systems.They noted, however, that the system maintained con-vective cells along the gust front for much longer pe-riods than the squall lines presented in WKR and Weis-man (1993) for similar low-level vertical wind shear,and noted further that the deeper-layer shear in this casemay have been critical for the strength and sustenanceof the simulated system. Evans and Doswell (2001)completed a climatological study of environments as-sociated with widespread convectively produced wind-storms, and also found that such events could occur forweaker magnitudes of low-level vertical wind shear thansuggested in previous simulations of WKR and Weis-man (1993).

The goal of the present paper is to revisit the RKWtheory that squall-line strength and longevity are es-pecially enhanced in the presence of moderate-to-stronglow-level vertical wind shear and that the process bywhich this happens is through shear-enhanced lifting bycool pools. To accomplish this goal, in the followingsection we revisit the primary building block of theRKW theory, that cold-pool lifting is especially en-hanced by low-level shear. This will be addressed bypresenting new results from a two-dimensional vortic-ity–streamfunction model of a density current spreadingin a sheared environment; numerical experiments aredone to demonstrate the effects on cold-pool lifting ofshallow versus deep shear, and of surface-based versuselevated shear. The new model also allows for a moredirect interpretation of this enhancement in terms of arelative balance between the circulation associated withthe cold pool and that associated with the shear, as orig-inally presented by RKW. In section 3, we will recon-sider the degree to which the cold-pool–shear interactionexplains the behavior of full, three-dimensional squall-line simulations, through the analysis of an extensiveset of new idealized simulations, with enhanced reso-lution on larger domains, which extend the results ofWKR to include both deeper and elevated shear layers.Section 4 contains further discussion and a summary.

2. Cold pools spreading in shear flow

At the time of the writing of RKW, there was onlyone paper in the literature explicitly recognizing thatambient vertical wind shear adds a fundamentally newelement to the dynamics of density currents (Jirka andArita 1987). Reasoning through the vorticity equation,RKW argued that deeper lifting should be expectedwhere the circulation of the cold pool is opposed bythat associated with the shear (cf. Fig. 1b). Shortly after


FIG. 3. (a) Initial condition for two-dimensional cold-pool-in-shearsimulations. Shaded region represents uniform block of cold air; hor-izontal lines indicate initial tracer field; definition of the initial shearlayer is indicated. (b) The three shear profiles used in the simulations.

RKW, a series of papers beginning with Xu (1992)looked at steady-flow solutions of gravity currents inambient shear contained in a channel of finite depth. Xu(1992) used overall mass and momentum balance as away of deducing the height of the gravity current at alarge distance from its leading edge h2`. Xu and Mon-crieff (1994), following the same methodology, considerthe effects of circulations within the cold pool on h2`.In both of these studies the height of the cold pool atthe leading edge is considered to be outside the realmof the theoretical calculations. Numerical simulationsby Xu et al. (1996) of gravity currents in an environmentof constant ambient shear showed the expected resultof deeper currents with shear than without. However asstated in of Xu et al. (1996, p. 771), ‘‘As in the [abovecited] theoretical models, the rigid upper boundary usedin our numerical study plays an important role in keep-ing the simulated density currents quasi steady . . .’’ Xueet al. (1997) extend the simulations of Xu et al. (1996)to the case of gravity currents with the ambient shearrestricted to low levels. Xue et al. (1997) acknowledgethat ‘‘Similar to the numerical experiments of RKW,these experiments show that the slope of the frontalinterface, the depth of the density current . . . are con-trolled by the shear of the low-level inflow.’’ However,the effects are analyzed in terms of the steady-flow mod-els, which as stated in Xu et al. (1996), are heavilyinfluenced by the upper rigid lid. In the following, wepresent calculations of cold pools spreading in shearbased on the vorticity–streamfunction version of theequations of motion. These calculations show that theRKW ideas, based on vorticity reasoning, are both qual-itatively and quantitatively accurate, and are not limitedto steady-state flows of the type considered in the above-cited work.

The formulation of the problem is motivated by thephenomenology depicted in Fig. 1a: rain from a decay-ing thunderstorm cell quickly chills and loads the airbelow it. Hence the fluid mechanical problem is to de-termine the way the cold air spreads at the ground andhow this spreading is affected by ambient shear. Tostudy this problem in a manner consistent with the ideasdeveloped in RKW, a simple two-dimensional vorticity–streamfunction model has been devised.

The governing equations for two-dimensional flow inthe x̃–z̃ plane are

dh̃2˜5 2] b̃ 1 ñ¹ h̃, (1)x̃dt̃

db̃2˜5 ñ¹ b̃, and (2)

dt̃

2¹̃ c̃ 5 h̃, (3)

where b̃ is the buoyancy, the ỹ component of vorticity,h̃the streamfunction, and the kinematic viscosity. Thec̃ ñ

material derivative d/dt 5 ] t̃ 1 ũ] x̃ 1 w̃] z̃, where thevelocity (ũ, w̃) 5 (] z̃ , 2] x̃ ). Here and throughout thisc̃ c̃

section all symbols with a tilde denote dimensional var-iables.

The basic flow to be analyzed is shown in Fig. 3a:there is at t̃ 5 0 a block of cold air described by thefunction

2b̃ , | x̃ | , l̃, z̃ , h̃0b̃ 5 (4)50, otherwisecontained within a channel of depth H̃ and length L̃.The initial vorticity (associated with the ambient shearflow shown in Fig. 3a; profiles used here are illustratedin Fig. 3b) is

h̃ , z̃ # z̃ # z̃0 1 2h̃ 5 (5)50, otherwisewhere 0 5 DŨ/(z̃2 2 z̃1); b̃0 and 0 are constants. Theh̃ h̃boundary conditions are w̃ 5 ] z̃ b̃ 5 ] z̃ 5 0 at z̃ 5 0,h̃H̃ and, for simplicity, periodicity at large distance x̃ 56L̃.

Dimensional analysis says that the solution can de-pend on seven nondimensional parameters, which weidentify as follows:

˜ ˜ ˜DU Dz̃ z̃ ñ l̃ H Lm, , , , , , , (6)h̃ h̃ h̃ h̃ h̃Ïb h̃ h̃Ïb h̃0 0

where D z̃ 5 z̃2 2 z̃1 and z̃m 5 (z̃2 1 z̃1)/2.Equations (1)–(5) are next nondimensionalized using

h̃ as the length scale, b̃0 as the buoyancy scale, Ïb h̃0


FIG. 4. Collapse of a cold pool with zero initial shear (DU 5 0)at (a) t 5 2 and (b) t 5 4. Here and subsequently, the shaded regiondenotes the area where the buoyancy b , 20.5; contours of stream-function c (interval 5 0.1) are indicated by the dashed lines; theheavy solid lines indicate the contours of the tracer field s (interval5 0.25).

as the velocity scale, and h̃/ as the time scale. TheÏb h̃0result is

dh25 2] b 1 n¹ h, (19)xdt

db25 n¹ b, (29)

dt2¹ c 5 h, (39)

21, |x | , l, z , 1b 5 (49)50, otherwise, and

h , z # z # z0 1 2h 5 (59)50, otherwise,where h0 5 DU/Dz, and n 5 /(h̃ ). The boundaryñ Ïb h̃0conditions are w 5 ]zb 5 ]zh 5 0 at z 5 0, H andperiodicity at x 5 6L. Symbols without a tilde are thenondimensional variables corresponding to the dimen-sional ones defined above. The model domain (x, z) ∈( 2L, 1L) 3 (0, H) is resolved with nx 3 nz 5 9603 80 uniform grid increments. Equations (19)–(39) arediscretized on a nonstaggered grid using second-order-accurate finite differencing in space and advanced intime through the leapfrog technique; the Poisson equa-tion (3) is solved using the National Center for Atmo-spheric Research (NCAR) library routine POIS.

The choices for the parameters in (6) are guided bythe following considerations. The cloud model simu-lations show that the cell-triggering lifting at the leadingedge of the surface outflow occurs perforce as soon asthe surface outflow is generated. Thus we focus here onthe evolution of the cold pool over short times choosingthe domain large enough so that it exerts no significantinfluence on the flow; here we take L 5 12, H 5 2 andwill present evidence that these are large enough. In theabsence of initial shear the problem described in Fig. 3is the so-called dam-break problem (e.g., Stoker 1957).Solutions of the shallow-water equations indicate that,after the dam breaks, the steady outflow from the res-ervoir is accompanied by a depression wave travelinginto the reservoir at speed . With two dams break-Ïb̃ h̃0ing (as in Fig. 3a), there will be a constant supply offluid to the outflow until the depression waves from bothdams meet, reflect, and travel back to the dam sites, thatis, until t 5 2l̃/h̃. We take l̃ 5 2h̃ and, guided by thedam-break problem, integrate Eqs. (19)–(39) over t ∈(0, 4) with a uniform time step of dt 5 0.01. It is alsoknown from past studies that a characteristic feature ofthe outflow is turbulent motion at its top. Thus, we taken 5 0.002, which is large enough to control the small-scale instability that occurs in the inviscid problem, butsmall enough to preserve the large-scale character ofthe solutions. With the latter four parameters of (6) spec-ified, we investigate the dependence of the solution onthe first three parameters in (6).

a. Density current-shear simulations

Consider first in Fig. 4 the collapse of the initial coldpool in the absence of ambient shear. The shaded regionrepresents values of b , 20.5 and the streamfunctionc is indicated in dashed lines with contour interval 50.1. To keep track of the vertical displacement of low-level air parcels, we define a tracer s such that it obeysthe same equation and boundary condition as b (29), butwith initial condition s(x, z, 0) 5 z. It is clear from (19)that vorticity of opposite signs is generated at the lateraledges of the initial cold pool; the generated vorticityimplies a flow by (39) that sinks and spreads the coldair while raising the surrounding low-level air parcels.[In all simulations the cold pool is well contained withinthe domain x ∈ (212, 112) at t 5 4.] By comparingthe value of s over the cold air with its undisturbedvalue and restricting attention to parcels for which s ,0.5, we have a precise measure of the maximum verticaldisplacement of low-level air parcels. Figure 5 showsdm [ {max(z 2 s); s , 0.5} ø 0.5 for DŨ/D z̃ 5 0 atthe simulation end time t 5 4 (case A).

Three distributions of ambient velocity are studiedhere (see Fig. 3b):

1) Surface-based, limited-depth shear layer

z 5 0.25, Dz 5 0.5, DU 5 0.25 to 1.5.m

2) Limited-depth shear layer centered at the initial cold-pool top

z 5 1.0, Dz 5 0.5, DU 5 0.25 to 1.5.m

3) Shear layer over the entire depth


FIG. 5. The maximum displacement (dm) of low-level (s , 0.5) airparcels at t 5 4 for all experiments. Letters indicate solutions dis-played in Figs. 4 and 6–9. The three curves correspond to the shearprofiles shown in Fig. 3b.

FIG. 6. As in Fig. 4 except DU 5 0.85, Dz 5 0.5, and zm 5 0.25.

FIG. 7. As in Fig. 4 except DU 5 1.5, Dz 5 0.5, and zm 5 0.25.

z 5 1.0, Dz 5 2.0, DU 5 1.0, 2.0, 3.0.m

The choice of DU in shear profile (iii) is made so thatDU/Dz is the same as it is in cases (i) and (ii) for DU5 0.25, 0.5, and 0.75.

The maximum low-level displacement dm at t 5 4 forall of these cases is shown in Fig. 5. For shear profile(i), there is clearly a shear that produces the maximumlift. Figure 6 shows the case with DU 5 0.85 (case B);the effect of the shear is evident at t 5 2 as it tends totilt the initial cold block downshear. However, by t 54 this tendency is opposed by the cold-pool circulationon the downshear edge and deep lifting occurs. Figure7 shows the solution for shear profile (i) with DU 51.50 (case C); in this case the downshear tilting of thecold pool is too strong for the cold-pool-generated cir-culation to overcome and the lifting is not as deep.

There has been some discussion in the literature onwhether the low-level shear is the most important featureof the shear distribution in determining the ability ofthe cold pool to lift air parcels (Xue 2000). Figure 5shows that with the shear layer displaced to the top ofthe original cold pool (ii), the effect on the low-levellifting is negligible. We have done experiments with theshear-layer place at intermediate levels (0.0 , zm , 0.5,Dz 5 0.5) and find a monotonic transition from no effecton lifting with shear profile (ii) to the strong effect as-sociated with shear profile (i). In a description of anelevated shear-layer numerical experiment, RKW (p.477) said that it is only the shear layer that ‘‘makescontact with’’ the cold pool that matters for enhancedlifting. The latter phrase is technically incorrect since,as just stated, the vorticity of even a marginally elevatedshear layer still has an associated circulation which actsat a distance [cf. Eq. (3)] to produce enhanced lifting.The phrase should have been ‘‘in close proximity to’’which would have made it consistent with the action-at-distance argument made in the same paragraph(RKW, p. 477; see the italicized phrase). Figure 8 showsthe simulation from shear profile (ii) with DU 5 0.85,indicating that there is no significant extra lifting of low-

level air beyond that obtained with zero ambient shear(Fig. 4) with the shear layer elevated completely abovethe cold pool.

Finally there have been a number of analytical andnumerical studies (e.g., Xu et al. 1996) which have ex-amined the case of uniform shear over the domain depth.Figure 5 shows that the lifting associated with a deepshear profile (iii) is similar to that of a surface-basedlimited-shear profile (ii) for DU/Dz , 1; however, be-yond that value the lifting is reduced, indicating thatdeep-layer shear is detrimental. Figure 9 shows the sim-ulation from (iii) with DU 5 2.0 (case E). It was alsonoted by Xu et al. (1996, p. 771) that the upper lid playsan important role in their solutions. The simulations


FIG. 8. As in Fig. 4 except DU 5 0.85, Dz 5 0.5, and zm 5 1.0.FIG. 9. As in Fig. 4 except DU 5 2.0, Dz 5 0.5, and zm 5 0.25.

discussed above were repeated with the vertical domainsize doubled (H 5 4) and there was no significant effecton the displacements shown in Fig. 5.

b. Vorticity dynamics

The displacement of an air parcel is defined by theequation

dd5 w. (7)

dt

Equation (7) says that an air parcel will experience morevertical displacement when it resides in a region of w. 0 for a longer time. A simple explanation based onvorticity dynamics [(19)–(39)] for the effect of shear onthe lifting ability of a spreading cold pool was given inRKW (sections 4a,b). With zero ambient shear, Fig. 4shows that lifted air is rapidly swept rearward of thecold-pool leading edge by the circulation associatedwith baroclinically generated vorticity; air parcels residein the updraft region for a relatively short time. Figure6 suggests that the circulation associated with the am-bient shear counters (on the downshear side) that as-sociated with the baroclinically generated vorticity and,consequently, air parcels reside in the updraft region fora relatively long time. With larger amounts of shear (Fig.7), fluid parcels spend less time in the updraft, and thevertical displacement is less again. A quantitative anal-ysis confirming the above ideas is shown in Fig. 10;Fig. 10a presents the full streamfunction (c) for the caseof maximum dm shown in Fig. 6. In Fig. 10b, the stream-function represents the flow associated only with h ,0 (the vorticity produced by the downshear cold pool)while Fig. 10c shows that associated only with h . 0(the mean shear vorticity plus the vorticity produced by

the upshear cold pool). The nearly balancing tendenciesof both on the downshear side of the cold pool is ap-parent.

c. Summary

Enhanced lifting at the downshear edge of the coldpool was identified by RKW as a major contribution tothe strength and longevity of squall lines. RKW’s orig-inal analysis was made in terms of the two-dimensionalequations of motion in the streamfunction–vorticityform. Here we have solved these equations numericallyto reconfirm the importance of the low-level shear andto show that the qualitative arguments of RKW arequantitatively accurate. The argument is that the deepestlifting occurs when the circulation associated with thecold pool is nearly balanced with that of the low-levelshear. By a simple control volume analysis RKW de-duced that this balance occurs when

˜ ÏDU 5 2b̃ h̃ [ C, (8)0 cwhere h̃c is the height of the cold pool. Examination ofFig. 6 shows that h̃c ø h̃/2, and so (8) gives /DŨÏb̃ h̃05 1, which is close to the experimentally determinedvalue of 0.85 (Fig. 5).

3. Squall-line simulations

Here we reconsider the degree to which the above-described concept of cold-pool–shear interaction is ap-plicable to understanding numerical simulations ofsquall lines. We first consider the basic influence ofshear on the overall structure of the squall lines, in termsof cellular characteristics and line-averaged system evo-lution. We then offer several possible measures of over-all squall-line strength to establish the degree to which


FIG. 10. Decomposition of the (a) flow field [c, contour interval(c.i.) 5 0.1] into that induced by the vorticity associated with the (b)downshear cold pool [c (h , 0), c.i. 5 0.05] and that induced bythe vorticity of the shear layer and the (c) upshear cold pool [c(h. 0), c.i. 5 0.05].

given environments could be considered more or lessfavorable. In view of the discussion given in the intro-duction, we consider the effects of shear depth and mag-nitude, as well as the solution sensitivity to the locationof the shear layer relative to the surface.

a. Experimental design

Squall lines are simulated herein using the Klemp–Wilhelmson (1978) cloud model within a domain 600km in the east–west (cross line) and 160 km in the north–south (along line) direction, with periodic conditionsspecified at the northern and southern boundaries. Gridspacing is 1 km in the horizontal direction and 500 min the vertical direction. The top of the domain is setat 17.5 km AGL, where a gravity wave radiation con-dition is applied. Such resolutions are considered suf-ficient for simulating convective system structure andevolution (e.g., Weisman et al. 1997; Adlerman andDroegemeier 2002), and are consistent with the manyrecent squall-line simulation studies in the literature thatare compared to the present results (e.g., Fovell andOgura 1989; Fovell and Tan 2000; Lafore and Moncrieff1989; Trier et al. 1997; Grabowski et al. 1998; Coniglioand Stensrud 2001). Only warm rain microphysics are

employed, and the Coriolis force is set to zero. Theexclusion of ice processes limits the interpretation ofcertain quantitative aspects of the present simulations,such as the extent and quantity of stratiform precipi-tation, and, to some degree, resultant cold-poolstrengths. However, past studies (e.g., Fovell and Ogura1988; Skamarock et al. 1994) suggest that these effectsdo not significantly impact the fundamental processesand relationships discussed herein. A free-slip conditionis applied at the surface, which could impact the inter-pretation of surface wind production in the simulations(e.g., Adlerman and Droegemeier 2002), but, again,there is no evidence that this simplification fundamen-tally alters the system-scale relationships being consid-ered.

The present experimental design is similar to that usedby WKR, with the exception of both finer resolution(previously 2-km horizontal and 750-m vertical gridspacing) and a much larger domain (600 km versus 160km) in the cross-line direction. We have found that theuse of such large cross-line dimensions is critical foraccurately representing the mature structure of the sim-ulated systems, as important circulation-forcing regionsat the rear of the system, associated with the cold poolat the surface and warm pool aloft, can be lost throughthe boundaries using smaller domains. In particular,Fovell and Ogura (1989) have noted that the degree ofdecay evident in the weaker shear simulations of RKWand WKR may have resulted from the constrained do-main sizes in those earlier simulations, rather than froman actual physical process. We have found that the useof 600-km cross-line domains is sufficient to avoid thisnumerical effect over the 6-h integration period.

The horizontally homogeneous initial state is char-acterized by a single profile of temperature and moisture(Fig. 11a), as previously used by WKR, which has aconvective available potential energy (CAPE) of 2200J kg21. This amount of CAPE is most applicable tostudying severe, midlatitude squall lines (e.g., Bluesteinand Jain 1985), but recent studies of tropical systems(e.g., Trier et al. 1997; Robe and Emanuel 2001) suggestthat the present relationships are applicable to tropicalthermodynamic environments as well. Eight basic uni-directional wind shear profiles are considered (Fig. 11b),the first four representing 2.5-, 5-, 7.5-, and 10-km-deepsurface-based shear layers, and the next three repre-senting the 2.5-, 5-, and 7.5-km-deep shear cases withthe shear layers now begining at 2.5 km AGL. Finally,for a few selected cases, the shear layer begins at 5 kmAGL, to accentuate the differences between the surface-based and elevated shear cases. The magnitude of thewind speed difference across the shear layers, Us, isvaried from 0 to 40 m s21, representing much of therange of environmental shear conditions associated withsquall lines.

The squall lines are initiated using a north–south-oriented line thermal of 1.5-K maximum potential tem-perature excess, with small (0.1 K) random potential


FIG. 11. (a) Thermodynamic sounding and (b) wind profiles usedfor simulations.

temperature perturbations added to the line thermal toallow for the development of three-dimensional featuresalong the line. This potential temperature perturbationwas the minimum necessary to trigger convection in areasonable amount of time (1 h) for the vast majorityof cases, and is small compared to the maximum po-tential temperature excess of 8 K for an undilute surfaceair parcel based on the environmental sounding (e.g.,Fig. 11a). It was necessary to increase the initial thermalperturbation to 2.0 K for many of the Us 5 30 m s21

cases, and 2.5 K for some of the Us 5 40 m s21 cases,in order to allow the initial convection to develop in asimilar amount of time compared to the weak- and mod-erate-shear simulations. However, this small differencein initiation procedure does not impact the basic con-clusions drawn herein.

b. Squall-line structure–shear dependencies

The simulated squall lines all evolve through thephases of evolution shown schematically in Fig. 2, asthe convectively generated cold pools strengthen over

time. However, the time period over which this evolu-tion takes place is inversely proportional to the overallmagnitude of the ambient shear, with the weakest-shearsimulations developing an overall upshear-tilted struc-ture within the first couple of hours and the strongest-shear simulations maintaining a predominantly down-shear-tilted structure throughout the 6-h integration pe-riod.

First, we investigate the role of surface-based envi-ronmental shear (e.g., Fig. 11b) on squall-line structure.The basic shear dependence is represented in Figs. 12and 13, which depict along-line averaged vertical crosssections and horizontal cross sections at 3 km AGL ofkey storm properties at 4 h for the Us 5 0, 10, 20, and30 m s21, 5-km-deep surface-based shear profiles, re-spectively. For the case with zero ambient shear (Figs.12a, 13a), the system structure at 4 h is highly disor-ganized, with scattered, short-lived, convective cells farbehind the leading edge of the spreading cold pool. WithUs 5 10 m s21 over 5 km AGL (Figs. 12b, 13b), thesimulated squall line is more organized, with a largernumber of weak convective cells scattered behind theleading edge of the surface cold pool, but it still exhibitsa predominantly upshear-tilted configuration, with anascending flow branch spreading predominantly rear-ward over a strong cold pool and divergent outflow atthe surface. As the ambient shear increases to Us 5 20m s21 over 5 km (Figs. 12c, 13c), the system-scale up-draft remains stronger and more upright, with the up-drafts and rain cells forming a more continuous linealong and just behind the leading edge of the surfacecold pool. Additionally, a more symmetric anvil outflowis evident aloft. Finally, for Us 5 30 m s21 over 5 km(Figs. 12d, 13d), the system-scale updraft remains up-right-to-downshear tilted through most of the simula-tion, with strong, highly three-dimensional cells, somewith supercellular characteristics, extending along theleading edge of the cold pool. A predominantly down-shear-tilted anvil is now evident aloft.

Figure 14 presents close-up vertical cross sections ofthe cold pools for these cases at representative locationsalong the line at 4 h. Consistent with the results for themore simple streamfunction model presented in section2 (e.g., Figs. 4, 6, 7), cold-pool depth (and strength)increases with increasing shear up to some optimal state(e.g., Us 5 20 m s21 for these simulations), but decreasesif the shear is increased even further. Of course, forthese more complete simulations, cold-pool depth,strength, and shear magnitude are not independent pa-rameters; for example, the subsequent strength of theconvection, which depends on the shear, influences thesubsequent strength/depth of the cold pool, etc. Still,we believe that the simple model offers important in-sights into the cold-pool behavior observed in the morecomplicated model.

The time series of maximum vertical velocity forthese cases (Fig. 15a) shows that significant updraftsare being produced out to 6 h for all shears considered,


FIG. 12. Line-averaged vertical cross section of system-relative flow vectors, negative buoyancyfield, and outline of the cloud field (thick line) for the (a) Us 5 0 m s21 (b) Us 5 10 m s21, (c)Us 5 20 m s21, and (d) Us 5 30 m s21 5-km-deep surface-based shear simulations at 4 h.Magnitudes of the buoyancy field between 20.01 and 20.1 m s22 are darkly stippled, withmagnitudes less than 20.1 m s22 lightly stippled. Vectors are included every 10 km in thehorizontal, and 500 m in the vertical, with a horizontal vector length of 10 km equal to vectormagnitude of 15 m s21. Tick marks are included every 10 km in the horizontal and 1 km in thevertical. Only a 240 km 3 14 km portion of the full domain is shown.

but also shows that the average maximum strength ofthe updrafts clearly increases for increasing magnitudesof shear, as reported in previous studies (e.g., Thorpeet al. 1982; WKR). This positive role of shear is evenmore pronounced in the 20-min averaged rain rates (Fig.16a), where the Us 5 20 and 30 m s21 rain rates farexceed those for the weaker shears. The regeneration ofsignificant updrafts through 6 h for the Us 5 0 and 10m s21 cases is a departure from the results of RKW andWKR, wherein the development of an upshear-tiltedstructure resulted in far more weakening of the subse-quent convection. As noted above, we believe that thisprevious result was an articraft of the numerical design,as simulations were run too long on domains that weretoo small. However, consistent with RKW and WKR,the new results shown in Figs. 12 and 13 indicate thatthe convection for the weaker shear cases is still widelyscattered behind the spreading cold pool, and is far lessorganized as compared to the behavior of the convectivecells in the stronger-shear cases. Consequently, the rainrates (Fig. 16a) are also far less for these weaker shears.

The relative importance of the depth of the ambientshear layer is demonstrated in Fig. 17 for the Us 5 20m s21 surface-based shear cases. When the shear layeris confined to the lowest 2.5 km AGL, the system struc-ture is similar to the 5-km-deep shear case. (cf. Fig. 12c

with Fig. 17a). Additionally, very strong bow-shapedupdraft segments at 3 km AGL are now evident alongthe line (Fig. 17b), with accompanying line-end vorticesand locally intense rear-inflow jets, as previously de-scribed by Weisman (1993). It is interesting that themaximum vertical velocities for the 2.5-km shear case(Fig. 15b, thick line) are actually weaker than for the5-km shear case (Fig. 15a, medium line), although therain rates are quite comparable (e.g., Fig. 16). As alsodiscussed by Weisman (1993), maximum vertical ve-locities are often weaker for such organized bow echoes,due to the interactions with the elevated rear-inflow jet,which forces the updraft current rearward above the jetlevel (usually 3–6 km AGL). However, updrafts are stillvery strong below this level, resulting in very large rainrates. When the shear layer is spread over the lowest7.5 km AGL (Fig. 17c), the system now maintains adownshear spreading anvil aloft, but a more upshear-tilted structure at lower levels, with a shallower surfacecold pool and a shallower front-to-rear ascending up-draft current. Individual updrafts and rain cells at 3 kmAGL are now scattered behind the leading edge of thesurface cold pool (Fig. 17d), and are much weaker anddisorganized than for the 2.5-km-depth shear case (Fig.13c). The rain rates are also significantly reduced fromthe shallower-shear cases (e.g., Fig. 16), although, again


FIG. 13. Horizontal cross section at 3 km AGL of system-relative flow vectors, rainwater mixingratio, and positive vertical velocity, for the (a) Us 5 0 m s21, (b) Us 5 10 m s21, (c) Us 5 20m s21, and (d) Us 5 30 m s21 5-km-deep surface-based shear simulations at 4 h. Vertical velocityis contoured at a 3 m s21 interval, with the rainwater lightly shaded for 0.001 to 0.004 g kg21

and darkly shaded for greater than 0.004 g kg21. The surface cold-pool boundary, defined as the21-K perturbation temperature contour, is depicted by the thick, dashed line. Vectors are includedevery 4 km in the horizontal, with a horizontal vector length of 4 km equal to a vector magnitudeof 20 m s21. Tick marks are included every 10 km. Only an 80 km 3 80 km portion of the fulldomain is shown.

maximum updrafts are comparable (e.g., Fig. 15). Thus,restricting a given net magnitude of shear (e.g., Us) toshallow layers produces a stronger squall line than ex-tending that shear over deeper layers.

A basic implication of the RKW theory is that am-bient shear confined to the depth of the cold pool ismost effective in countering the cold-pool circulationand thus promoting deeper cold-pool lifting. As shownin section 2 for a simple density current, if the shearlayer is only above the top of the cold pool, the amountof lifting produced is significantly reduced (e.g., Fig.5). Based on this, one would expect squall lines to bestronger and more long-lived if the shear layer weresurface-based rather than elevated. Indeed, if the shearlayer is located high enough, there would be virtuallyno interaction with the cold pool, and thus the overallsystem structure would revert more closely to the case

with weak vertical wind shear. The effects of elevatingthe shear layer for the Us 5 20 m s21 5-km-deep shearcase is presented in Fig. 18, where the shear layer isfirst located from 2.5 to 7.5 km, and then from 5.0 to10.0 km AGL. As the shear layer is lifted, the systemmaintains more of a downshear-tilted anvil aloft, in re-sponse to the upper-level shear, but the low-level cir-culation becomes more upshear-tilted, with the convec-tive cells becoming more scattered and located fartherbehind a shallower spreading surface cold pool. Rainrates are also far reduced from the comparable surface-based shear cases (e.g., Fig. 16). Thus, the overall sys-tem structure approximately reverts back to that asso-ciated with much weaker surface-based wind shears (cf.Figs. 18c,d with Figs. 12a, 13a).

As mentioned in the introduction, it has been pro-posed that the effect of shear (deep or shallow) is to


FIG. 14. Representative vertical cross section of system-relativeflow vectors and negative buoyancy field through the cold pool forthe (a) Us 5 0 m s21, (b) Us 5 10 m s21, (c) Us 5 20 m s21, and(d) Us 5 30 m s21 5-km-deep surface-based shear simulationsat 4 h. Magnitudes of the buoyancy field between 20.015 and

←

20.15 m s22 are lightly stippled, with magnitudes less than 20.15m s22 darkly stippled. Vectors are included every 2 km in the hori-zontal, and 500 m in the vertical, with a horizontal vector length of2 km equal to vector magnitude of 15 m s21. Tick marks are includedevery 1 km in the horizontal and 500 m in the vertical. Only a 40km 3 4 km portion of the full domain is shown.

allow individual cells to move with the surface coldpool, thus enhancing cell strength and lifetime (e.g.,Lafore and Moncrieff 1989; Thorpe et al. 1982; Mon-crieff and Liu 1999). From this perspective, the low-level-shear–cold-pool interactions of RKW are not crit-ical for system strength and longevity. Indeed, elevatingthe shear to 2.5 km AGL for the Us 5 20 m s21 5-km-deep shear simulation does produce an enhancement insystem strength and organization compared to the zeroshear simulation, although not to the extent observedwhen the shear is surface based. Some of this enhance-ment may be due to the influence that the circulationassociated with the elevated shear may still be havingon the circulation of the cold pool, as discussed in sec-tion 2a. Still, it is worth considering more carefully therole of the elevated-layer shear in maintaining or re-generating convective cells within such a system.

For the zero-shear case, new, weak cells are primarilyinitiated about 20 km behind the leading edge of thecold pool (e.g., Fig. 13a) by the collision of outflowsfrom preexisting cells, which locally deepen the coldpool, thereby enhancing the lifting of the ambient airflowing rearward above the cold pool. The same processalso contributes to cell regeneration somewhat for theUs 5 20 m s21 elevated shear case. However, we findthat lifting above the cold pool is additionally enhancedby the interactions of the elevated shear with the neg-ative buoyancy regions of the preexisting raincells, asalso described for low-level-shear–cold-pool interac-tions. This process is demonstrated in Fig. 19, whichpresents the early life cycle of a new convective elementwithin the elevated shear-layer case shown in Figs.18a,b. Initially, a preexisting raincell and associateddowndraft are located about 20 km behind the cold pool(Fig. 19a). Air rises abruptly at the leading edge of thecold pool and then proceeds rearward for 10 to 20 kmwithout any further significant ascent, consistent withthe cold-pool circulation overwhelming the ambientlow-level shear. Alternately, the lack of ascent at thisstage can be attributed to downward-directed nonhy-drostatic pressure gradient forcing associated with thecold-pool circulation, which counters the weak positivebuoyancy forcing achieved through the lifting (e.g.,Fovell and Tan 2000; Trier et al. 1997; Weisman et al.1988). As the air parcels travel rearward, they encountera negative buoyancy region that extends at least 3 kmabove the height of the cold pool, associated with thedecaying raincell. Buoyancy in the cloud model is de-fined as


FIG. 15. Time series of maximum vertical velocity for the (a) Us5 0, 10, 20, and 30 m s21 over 5-km-deep surface-based shear sim-ulations; and (b) Us 5 20 m s21 over 2.5 km, Us 5 20 m s21 over7.5 km, Us 5 20 m s21 over 2.5–7.5 km, and Us 5 20 m s21 over5.0–10.0-km-deep shear simulations.

FIG. 16. Time series of rainfall rates [31010 kg (20 min)21] for theUs 5 0, 10, 20, and 30 m s21 over 5-km-deep surface-based shearsimulations; and (b) Us 5 20 m s21 over 2.5 km, Us 5 20 m s21 over7.5 km, Us 5 20 m s21 over 2.5–7.5 km, and Us 5 20 m s21 over5.0–10.0-km-deep shear simulations.

u9B [ g 1 0.61(q 2 q ) 2 q 2 q , (9)y y c r[ ]u

where is the mean potential temperature; u9 is theuperturbation potential temperature; and qy, qc, and qrare the mixing ratios of water vapor, cloud water, andrainwater, respectively. This elevated negative buoyancyregion is primarily associated with rain loading 2gqr,while at lower levels the negative buoyancy is primarilydue to negative potential temperature perturbations re-sulting from evaporation. As the air parcels encounterthis region, the interaction of the elevated shear withthe positive horizontal buoyancy gradient enhances thelifting just ahead of the decaying raincell (Fig. 19b),leading to the subsequent development of a new con-vective cell (Fig. 19c).

This proposed process of enhanced cell regenerationfor the elevated shear cases may not be the only processcontributing in these cases, the details of which are be-yond the scope of the present paper. In any event, our

examination of new cell growth in all of these simu-lations shows no evidence that a preexisting convectiveupdraft moving with the cold pool contributes to thecell regeneration process, as hypothesized in previousstudies. In fact, the region of cell regeneration shownin Fig. 19 is characterized by downdraft prior to newcell growth (Fig. 19a). Further discussion of cell lifecycles in simulated squall lines is presented in Fovelland Tan (2000).

The results for the full range of simulations are sum-marized in Table 1a, where the overall system-scalestructure between 3 and 6 h is subjectively classified aseither an upshear-tilted, multicellular system with cellsscattered behind the leading edge of the cold pool (UP),a predominantly upright system with linear or bow-shaped segments of cells just behind the leading edgeof the cold pool (B), or a system composed of moreisolated, upright-to-downshear-tilted cells or embeddedsupercells right above the leading edge of the cold pool(D), as described in the earlier examples. It should be


FIG. 17. Line-averaged vertical cross sections and horizontal cross sections at 3 km AGL at 4h for the (a), (b) Us 5 20 m s21 2.5-km-deep and (c), (d) Us 5 20 m s21 7.5-km-deep surface-based shear simulations, as described for Figs. 12 and 13, respectively.

noted that the cases classified as either B or D includesignificant three-dimensional features embedded withinthe lines that could potentially limit the applicability oftwo-dimensional theories to interpreting overall systemcharacteristics. This will also significantly impact com-parisons with two-dimensional numerical models forthis range of environments, as structures such as bowechoes and supercells cannot be reproduced in suchmodels. Indeed, the inclusion of the third dimensionreleases the strong-shear constraint on squall-linestrength noted in the two-dimensional studies of Hane(1973), Thorpe et al. (1982), and discussed by Lilly(1979, 1986), allowing for the development of long-lived convective structures that can gain energy fromthe shear. Despite such potential complications, a sys-tematic change/increase in organization, from upshearto upright to downshear, for increasing magnitude ofshear, is evident for all shear depth/location categoriesconsidered, consistent with the simple two-dimensionalconcepts suggested in Fig. 2. Additionally, more netshear (e.g., Us) is needed to maintain an upright- ordownshear-tilted system for the elevated versus surface-

based shear cases, suggesting a reduced role of the el-evated shear in contributing to system organization. Fi-nally, well-organized bow echoes, as characterized inFig. 17b, are most prevalent for the strong, shallow,surface-based shears, as found in Weisman (1993), al-though for slightly weaker shears than found in thisprevious study (e.g., bow echoes occur for Us 5 15 ms21 over 2.5 km in the present study, while Us 5 20 ms21 over 2.5 km was necessary in the previous study).These small differences are most likely related to boththe finer resolutions and longer integration times usedin the present study.

RKW and WKR classify overall squall-line strengthand structure in terms of the parameter, C/Du, where Crepresents the theoretical speed of propagation of a two-dimensional gravity current (e.g., cold pool; Benjamin1968), given by

H

2C 5 2 (2B) dz, (10)E0

where H represents the depth of the cold pool, B rep-


FIG. 18. Line-averaged vertical cross sections and horizontal cross sections at 3 km AGL at 4h for the (a), (b) Us 5 20 m s21 2.5–7.5-km-deep and (c), (d) Us 5 20 m s21 5.0–10.0-km-deepelevated shear simulations, as described in Figs. 12 and 13, respectively.

resents the buoyancy defined in (9), and Du is the am-bient shear, given as the difference in line-perpendicularambient wind between the surface and a level compa-rable to the depth of the cold pool. From the RKWperspective, a squall line takes on an overall upshear,upright, or downshear configuration for magnitudes ofC/Du greater than, equal to, or less than 1, respectively,as depicted in Fig. 2.

Because of the critical importance of the cold poolin contributing to the structure and evolution of squalllines, we first present in Table 1b calculations of Crepresentative of the mature phase of each of the sim-ulations. For this purpose, the buoyancy field, B, is av-eraged along the line, and a representative cold-pooldepth, H (defined as the level at which B vanishes), andminimum buoyancy, Bmin, is estimated for the 20-kmregion extending behind the leading edge of the coldpool. The buoyancy field is then assumed to decreaselinearly to zero at height H, allowing for the followingsimplification of (10):

2C 5 2B 3 H.min (11)

For the present purposes, C is calculated using (11) at3, 4, and 5 h and then averaged over these three timesto represent the average conditions observed during themature phase of the present simulations; the standarddeviation of C from the calculated average is small (1–2 m s21) over the time period included.

Table 1b reveals a maximum in resultant cold-poolstrength for the Us 5 15–25 m s21, 2.5-km-deep surface-based shear cases. Here, C decreases for both weakeror stronger shears and also decreases for increasingshear depth. This enhancement in cold-pool strength forthe shallowest shear examples is most directly relatedto an increase in cold-pool depth. A particularly inter-esting result is that C is significantly reduced for all ofthe elevated shear simulations. Analysis shows that thecold pools for the elevated shear cases are also char-acterized by higher values of ue, suggesting that thepotentially coldest midlevel air is not reaching the sur-face as readily as for the surface-based shear simula-tions. Thus, the decrease in cold-pool strength for theseelevated shear cases may be more related to the negative


FIG. 19. Vertical cross sections of flow vectors, buoyancy, andrainwater at (a) 3 h, 20 min; (b) 3 h, 25 min; and (c) 3 h, 30 minthrough a developing convective cell for the Us 5 20 m s21 2.5–7.5-km-deep elevated shear simulation. Buoyancy is contoured at 0.02m s22 intervals, with rainwater mixing ratios greater than 0.001 gkg21 shaded. Vectors are presented every other grid point in thehorizontal direction, with a vector length of one grid point equivalentto a vector magnitude of 10 m s21.

influence of midlevel shear on the development of thesystem-scale downdrafts, rather than a direct conse-quence of cold-pool–shear interactions. The large var-iation in C among these simulations, all using the samethermodynamic sounding, underscores the complicatedinterdependence of this parameter with other aspects ofsystem structure (e.g., RKW; Weisman 1992).

In Table 1c C/Du is presented for all of the simula-tions. In RKW and WKR, C/Du is calculated using a2.5-km depth for the shear calculation. However, for the

present simulations, a better correspondence with over-all squall-line structure was obtained with Du calculatedover the lowest 5 km AGL, reflecting the ability of theshear in the 2.5–5-km layer to still contribute to cellregeneration somewhat, as shown in Fig. 19 (C/Du ascalculated using a 2.5-km depth can be derived fromTable 1c by multiplying the 5-, 7.5-, and 10-km-deepsurface-based shear ratios by 2, etc.). Within this con-text, upshear-tilted systems generally occur for magni-tudes of C/Du greater than 1.5 (averaged along thesquall line between 3 and 5 h) with predominantlydownshear-tilted systems occurring for C/Du less than1. The most upright convective systems are found forC/Du between 1 and 1.5. As discussed by Lafore andMoncrieff (1989) and Weisman (1992), rear-inflow jetsbecome a significant component of the system-scale cir-culation during the mature phase of many of these sim-ulated squall lines, potentially complicating the simpleC/Du relationships being espoused. In particular, Weis-man (1992) showed that such rear-inflow jet could beincorporated into the present theory by including thevertical shear of the jet within the cold pool in the bal-ance condition; for example, the appropriate ratio wouldnow become C/(Du 1 Duj), where Duj represents thevertical shear within the cold pool. The simpler C/Durelationship, however, still provides a useful guide ofoverall system structure for a wide range of shear en-vironments, emphasizing the general applicability of theoriginal RKW concepts.

One of the concerns stated in Lafore and Moncrieff(1989) as to the application of C/Du concepts was thatDu may change significantly over time, as the convec-tion begins to modify the ‘‘environmental’’ shear profileahead of the system. Indeed, their Fig. 4a depicts sig-nificant modifications, especially above 6 km AGL,within such a two-dimensional squall-line simulation.This issue is highlighted further by Garner and Thorpe(1992), who, using a highly parameterized two-dimen-sional squall-line model, found even more substantialdownstream modifications to the shear, extending to lowlevels as well. In order to address this concern for thepresent simulations, we computed vertical profiles of Uaveraged over an 80-km region ahead of the surfacegust front, and also averaged along the line at 4 h (Fig.20), for the Us 5 0, 10, 20, and 30 m s21 over 5-kmshear simulations discussed in Figs. 12, 13, and 14. Asdiscussed by Lafore and Moncrieff (1989), the presentresults also show significant modifications of the shearprofile above 6 km, primarily reflecting the upper-leveldownshear outflow circulation, which maximizes be-tween 10 and 12 km AGL. However, the modificationsto the shear are less than 1 m s21 over the lowest 3.0km AGL, with generally less than a 3 m s21 modificationbetween 3 and 6 km AGL for all the cases considered.Since the dependence of squall-line structure on shearis heavily weighted to the shear over the lowest 5 kmAGL, and especially the lowest 2.5 km AGL, these en-vironmental shear modifications will not significantly


TABLE 1a. System structure 3–6 h. UP: Upshear multicellular, B: Bow echo, and D: Downshear cells, supercells.

Us (m s21)

Surface-based shear

2.5 km 5 km 7.5 km 10 km

Elevated shear (2.5 km)

2.5 km 5 km 7.5 km

(5 km)

5 km

0101520

UPUP

B/UPB

UPUP

UP/B

UPUPUP

UPUPUP

UPUP

B/UP

UPUPUP

UPUPUP

UP

UP253040

BD/B

D/BD/BD/B

DDD

UPD/UP

D

D/BD/B

D/UPDD

UPD/UP

DUP

TABLE 1b. Average C (m s21) (3–5 h).

Us (m s21)

Surface-based shear

2.5 km 5 km 7.5 km 10 km


2.5 km 5 km 7.5 km

(5 km)

5 km

0101520

19.023.025.026.0

21.021.525.0

20.019.520.0

20.019.520.0

19.019.018.0

20.019.018.5

19.019.019.0

18.5

18.0253040

25.021.0

23.021.018.0

19.020.019.0

19.019.520.5

19.517.5

18.018.518.0

19.017.519.0

17.5

affect the C/Du relationships discussed above. Theseadjustments to the wind profile on the downshear sideof the squall line are consistent with the theoretical ar-guments and numerical calculations of Fovell (2002),who also clarified that the more substantial modifica-tions noted by Garner and Thorpe (1992) were an ar-tifact of their simplified moisture parameterizationscheme.

c. An ‘‘optimal’’ state

One of the primary goals of this paper is to revisitthe concept of an optimal environment for squall-linestrength and longevity. For the case of a cold poolspreading in a sheared flow, optimality is defined simplyas the environment that produces the deepest, most up-right lifting at the leading edge of the cold pool. Fromthis perspective, an optimal state is envisioned whenC/Du, as presented in Table 1c, is close to 1. However,for full squall lines for which buoyancy is generatedwithin the updrafts, and for which the resulting precip-itating convective cells can have important feedbackson the evolution of the cold pool, etc., the concept ofoptimality is not as clear-cut. What is optimal for theproduction of individually strong updrafts may not bethe same as what is optimal for the production of strongdowndrafts and surface winds, or surface precipitation.For example, a strongly sheared environment might beconducive to the development of a few strong, isolatedsupercells embedded within the line, but a more weaklysheared environment might be more conducive to thedevelopment of a more solid line of convective cellsand overall heavier rainfall. It is the ability to producea more solid line of convective cells rather than super-

cells that is specifically addressed by RKW. In order toaddress the variety of ways in which system optimalitycould be defined, Table 2 presents several commonlyused parameters that measure various aspects of systemstrength.

Table 2a presents the total rainfall observed between1 and 6 h for each simulation. The first hour is neglectedhere to avoid some of the differences due to the timingand strength of the initial sequence of cells among thevarious environments. Total rainfall increases signifi-cantly as the net shear (Us) increases in magnitude forthe 2.5- and 5-km surface-based shear simulations.However, total rainfall decreases significantly when thenet surface-based shear (Us) is spread over 7.5 or 10km. Rainfall also decreases markedly for all of the el-evated shear cases. All in all, the most optimal squalllines from this perspective are found for magnitudes ofsurface-based shear between Us 5 20 and 30 m s21,with the shear layer confined to the lowest 2.5 or 5 kmAGL. Indeed, an ‘‘optimization’’ for the moderate sur-face-based shears would have even been more evidenthad we not increased the initial potential temperatureperturbation to initiate convection for many of the Us5 30 m s21 or greater shear cases, better reflecting theultimately negative influence of such strong shears onsystem development.

These results can also be interpreted to consider theeffects of extending a layer of a given magnitude ofshear over a deeper depth. For example, rainfall ratesincrease dramatically as a surface-based shear of Us 510 m s21 over 2.5 km is deepened to Us 5 20 m s21

over 5 km, but then decrease somewhat if the shear layeris deepened further to Us 5 30 m s21 over 7.5 km orUs 5 40 m s21 over 10 km. This result can also be


TABLE 1c. Average C/DUs (3–5 h).

Us (m s21)

Surface-based shear

2.5 km 5 km 7.5 km 10 km


2.5 km 5 km 7.5 km

(5 km)

5 km

0101520

—2.31.71.3

2.11.41.3

3.02.01.5

4.02.62.0

1.91.30.9

4.02.51.9

5.83.82.8

—

—253040

1.00.7

0.90.70.5

1.11.00.7

1.51.31.0

0.80.6

1.41.20.9

2.31.81.5

—

FIG. 20. Vertical profiles of the U component of the wind averagedalong the line over an 80-km region ahead of the squall lines, for the(a)–(d) Us 5 0, 10, 20, and 30 m s21 5-km surface-based shearsimulations, respectively.

interpreted from the C/Du perspective presented in Table1c. Starting with a suboptimal state (C/Du 5 2.3), deep-ening the shear layer produces a nearly optimal state(C/Du 5 1.3), but further deepening does not produceany additional benefit (C/Du 5 1). Similarly, if we ex-tend the already nearly optimal Us 5 20 m s21 over the2.5-km shear case (C/Du 5 1.3) to a shear magnitudeof Us 5 40 m s21 over 5 km (C/Du 5 0.5), only slightenhancement in total rainfall is observed. From a two-dimensional perspective, one would expect this lattercase with C/Du 5 0.5 to be weaker due to the excep-tionally strong shear (indeed, the initial temperature per-turbation had to be increased from 1.5 to 2 K to allowthe convection for this case even to initiate). However,as previously discussed, the ability to develop more iso-lated, three-dimensional cells or supercells (given con-vective initiation) can mitigate the negative influence ofthe stronger shears.

As a similar integrated measure of system strength,

Table 2b presents the total condensation observed from1 through 6 h for each simulation, which was one ofthe primary measures applied in previous study ofWKR. This parameter indicates a similar increase inoverall system strength as the 2.5- or 5-km-deep surface-based shears increase from Us 5 0 to 20 m s21. As alsoshown for the total rainfall, total condensation decreasesfor either deeper or elevated shear layers, with virtuallyno enhancement beyond the zero-shear case evident ifthe shear layer is lifted to above 5 km AGL. Theseresults differ somewhat from WKR, which showed adramatic decrease in total condensation as the sheardepth increased from 2.5 to 5 km AGL. However, asdiscussed above, these differences are directly attrib-utable to the use of much smaller domains and coarserresolutions in WKR that restricted the ability to regen-erate convective cells in the deeper shears and also togenerate correctly system-scale features, such as rear-inflow jets.

As a measure of the ability of the given environmentto produce strong, embedded convective cells over along period of time, Table 2c presents the maximumvertical velocity observed between 3 and 6 h for eachsimulation. Like the previous parameters, maximumvertical velocity increases somewhat with increasingshear magnitude, but unlike the previous parameters,maximum vertical velocity also increases with sheardepth and elevation. This result is most related to thetendency to produce more isolated convective cells forthe deeper or elevated shear cases. Such isolated cells,which are more inherently three-dimensional, can bestronger than the more linelike updraft features evidentfor the shallower shears. Perhaps most surprisingly, thestrongest individual cells occurred with the strong, el-evated shear cases. These simulations were additionallycharacterized by weaker cold pools (e.g., Table 1b).Coniglio and Stensrud (2001) similarly found that theaddition of upper-level shear could enhance longer-termupdraft strength for idealized two-dimensional simula-tions. Hence, maximum updraft strength offers a sig-nificantly different perspective of optimality than thepreviously discussed parameters.

Finally, recent modeling and observational studieshave also emphasized the potential for such convectivesystems to produce strong surface winds over long pe-


TABLE 2a. Total rainfall (1–6 h).

Us (m s21)

Surface-based shear

2.5 km 5 km 7.5 km 10 km


2.5 km 5 km 7.5 km

(5 km)

5 km

0101520

0.440.590.710.83

0.560.730.86

0.510.610.68

0.490.550.61

0.520.680.60

0.500.550.60

0.480.510.56

0.46

0.46253040

0.860.87

0.890.910.87

0.730.790.87

0.660.700.80

0.770.71

0.640.670.70

0.590.600.66

0.47

TABLE 2b. Total condensation (1–6 h).

Us (m s21)

Surface-based shear

2.5 km 5 km 7.5 km 10 km


2.5 km 5 km 7.5 km

(5 km)

5 km

0101520

1.161.201.291.37

1.181.341.45

1.111.171.20

1.111.131.17

1.121.271.20

1.121.081.12

1.151.111.12

1.15

1.12253040

1.401.38

1.461.411.41

1.181.211.33

1.161.141.21

1.261.17

1.091.081.15

1.111.081.11

1.07

riods of time (e.g., Weisman 1993; Coniglio and Stens-rud 2001; Evans and Doswell 2001). Long-lived, severewind-producing systems have often been associatedwith bow-echo-type structures embedded within largerconvective systems. Table 2d presents both the average(averaged along the line and over time) and maximumU (east–west) component of the ground-relative windat 250 m (the lowest grid level), based on the output at3, 4, and 5 h, for each of the simulations. These resultsshow that a broad range of shear conditions potentiallysupport strong surface-wind production in these simu-lations, but the strongest average winds are generallyproduced for the 2.5–5-km surface-based shear cases,for shear magnitudes between Us 5 10 and 25 m s21.As shown in Table 1b these cases are also characterizedby the strongest and deepest line-averaged cold pools,suggesting that the horizontal accelerations associatedwith the hydrostatic pressure excesses within the coldpools are significant contributors to strong surface-windproduction in these simulations. This shear regime alsoincludes as a subset the environments conducive to thedevelopment of well-organized bow echoes within thepresent simulations (e.g., Table 1a). However, poten-tially strong surface-wind-producing systems are not re-stricted to just such bow echo cases, but extend to moreweakly and strongly sheared environments as well, asalso noted in the previous simulations of Weisman(1993). It is also important to note that surface windsin both idealized simulations and observations increasein magnitude for increasing magnitudes of CAPE (e.g.,Weisman 1993; Evans and Doswell 2001; Bluestein andJain 1985; Bluestein et al. 1987), although this param-eter is not varied in the present study.

Comparisons with more recent modeling and obser-

vational studies of severe wind-producing systems showmuch consistency with the present results. The Coniglioand Stensrud (2001) composite environment for a de-recho event includes a magnitude of shear of 14–16 ms21 over the lowest 5 km AGL, with much of this shearresiding in the lowest 3 km, with an additional 15 ms21 of shear between 5 and 10 km. Indeed, such anenvironment is well within the range of shears that pro-duce long-lived swaths of potentially severe surfacewinds (defined here as maximum winds greater than 30m s21 at 250 m AGL) within the present simulations.Magnitudes of C/Du for their simulation (with Du cal-culated over a 2.5-km depth) averaged about 2 duringthe mature phase of the system, and the overall systemstructure was composed of upshear-tilted updrafts withsome embedded bow echoes. This is also reasonablyconsistent with the present simulations for which 15 ms21 of shear over the lowest 2.5–5 km AGL representsa transitional regime between the more disorganized,upshear-tilted systems and more upright systems withembedded bow echoes. However, the present simula-tions also emphasize that stronger shears, especiallywhen confined to low levels, produce even stronger,more coherent bow-echo-type structures and potentiallyeven stronger maximum surface winds than the envi-ronment considered by Coniglio and Stensrud. Evansand Doswell (2001) found that 75% of the derechoevents in their climatology had 0–6 km AGL shear mag-nitudes of greater than 11.6 m s21, with 25% of thecases having magnitudes greater than 20 m s21, alsoreasonably consistent with the present results. Theirstudy, however, did not specifically address the asso-ciation between shear and attendant convecctive orga-nization.


TABLE 2c. Wmax (m s21) (3–6 h).

Us (m s21)

Surface-based shear

2.5 km 5 km 7.5 km 10 km


2.5 km 5 km 7.5 km

(5 km)

5 km

0101520

27302929

303532

293436

293339

313640

333739

313439

32

33253040

3939

353838

373835

383838

4140

424038

434241

42

TABLE 2d. Avg/Max surface U (m s21) (3–5 h).

Us (m s21)

Surface-based shear

2.5 km 5 km 7.5 km 10 km


2.5 km 5 km 7.5 km

(5 km)

5 km

0101520

14/2321/3022/3622/37

20/3022/3222/35

18/2919/3320/37

17/2820/3220/34

17/3618/3816/40

16/2916/3116/35

15/2617/2816/33

14/30

16/28253040

21/4016/41

19/4214/3514/29

18/3312/3510/23

20/3517/3313/38

13/3710/35

13/3111/3411/25

17/3115/3314/34

17/31

4. Summary and discussion

In the foregoing, we have addressed issues in therecent literature concerning the role of ambient verticalwind shear in controlling the structure, strength, andlongevity of squall lines. A particular concern has beenwhether an appropriate balance between the cold-pool-generated circulation and the circulation associated withlow-level ambient shear is a necessary requirement forsustaining most strong, long-lived squall lines, as sug-gested by RKW, WKR, and Rotunno et al. (1990), orwhether other aspects of the shear profile, such as deeperor elevated shear layers could be as important in con-trolling system structure, strength, and longevity.

In order to address these concerns, we first revisitedthe basic premise of RKW theory, that the lifting pro-duced at the leading edge of a spreading cold pool (den-sity current) is ‘‘optimized’’ when the cold-pool cir-culation is countered by the opposite circulation asso-ciated with the ambient low-level vertical wind shear.A series of idealized density current simulations with atwo-dimensional vorticity–streamfunction model werepresented for a variety of shear magnitudes, depths, andelevations that reconfirm this basic idea, and add furthersupport to the vorticity-balance interpretations origi-nally presented by RKW.

In order to address the applicability of these conceptsto interpreting full-physics cloud-model simulations ofsquall lines, we presented analyses from a large seriesof new idealized squall-line simulations with larger do-mains, higher resolution, and a wider range of environ-mental shear conditions than originally presented byRKW and WKR, including both deeper and elevatedshear layers. As also discussed by Fovell and Ogura(1989), the use of large cross-line domains was found

to be critical for accurately representing the mature sys-tem-scale structure. In particular, the abrupt demise ofthe weaker-shear, suboptimal squall lines in the moreconstrained-domain simulations of RKW and WKR thusappear to be more of a numerical artifact than a physicaleffect. This does not impact the overall influence ofshear on system strength and structure as described inthose previous papers. However, squall-line longevity,without reference to system strength and structure, isnot found to be as sensitive to shear as in the paststudies.

Using several different measures of system strength,the present results reconfirm that squall-line strengthover a 6-h period is enhanced when moderate-to-strongshear is restricted to low levels, as found by Thorpe etal. (1982), RKW, and WKR, but now also extend therange of shear depths conducive to such enhancementfrom 2.5 to 5 km AGL. Increasing the surface-basedshear depth to 7.5 or 10 km allowed for the developmentof strong, isolated convective cells along the line in theregime of moderate-to-strong shear, but such systemswere weaker in terms of overall condensation and rain-fall output. Locating the shear layer above 2.5 km AGLalso allowed for significant, long-lived systems, al-though weaker and less organized than for the surface-based shears. Locating the shear layer above 5 km AGL,however, resulted in a much weaker and less-organizedsystem, similar to the system obtained with no verticalwind shear.

It is also shown here that the ratio C/Du (with Ducalculated over the lowest 5 km AGL) offers a usefulguide of overall squall-line structure for much of therange of shears considered, as previously presented byRKW and WKR, with predominantly upshear-tilted, up-


right, or downshear-tilted structures occurring for mag-nitudes of C/Du greater than 1.5, 1–1.5, or less than 1,respectively. In terms of defining an optimal state forsquall-line strength over the 6-h period based on a mag-nitude of C/Du approaching 1, it is clear that such adetermination is open to reasonable differences of in-terpretation, depending largely on the parameters chosenas measures of such system strength. However, whenconsidering the parameters that most nearly address theoverall two-dimensional character of the convective sys-tems (e.g., total rainfall, average surface wind, etc.),which is the true intention of the RKW approach, thereis a strong degree of correspondence between the presentcases that are more optimal from the C/Du perspective,and the tendency for the system to be composed of moreerect, leading-line convection.

Some have interpreted the results of RKW and WKRas stating that a balance between the cold pool andambient shear is ‘‘necessary’’ to support a strong, long-lived squall line. This seems inconsistent with the factthat significant, long-lived squall lines are frequentlyobserved in weaker, suboptimal shear environmentsfrom the RKW perspective (e.g., C/Du greater than 1;Coniglio and Stensrud 2001; Evans and Doswell 2001).However, as stated in Rotunno et al. (1990), ‘‘We rec-ognize that systems that do not achieve an appropriatebalance between the low-level shear and the cold poolstrength may last a long time; however, we believe thatin most instances, they will be weaker and shorter-livedthan if the shear were in balance with the cold pool.’’We feel that the present expanded results add supportto this conclusion. Significant, long-lived systems areproduced over a wider range of environments than strict-ly ‘‘optimal,’’ but squall-line strength is clearly en-hanced when moderate-to strong surface-based shear isconfined to the lowest 2.5–5 km AGL. Perhaps the moreimportant point, though, is that the relative strength ofthe cold pool and ambient shear has a profound effecton system organization over the entire range of envi-ronments considered, whether optimal or not, and, thus,represents a highly useful concept for describing overallsystem properties.

The present results were derived for an infinitely longsquall line that was constrained to be perpendicular tothe ambient shear in a thermodynamic environmentmost representative of severe, midlatitude squall lines.The same shear dependencies, however, can be derivedby starting with isolated or short lines of convectivecells, in both tropical and midlatitude environmentsalike. Starting in a horizontally homogeneous environ-ment, such systems invariably reorient themselves to beperpendicular to the shear, through the continual regen-eration of new cells along the cold pool, and, as withthe present results, such regeneration is especially en-hanced when the shear is confined to low levels (e.g.,Weisman and Klemp 1986; Weisman 1993; Robe andEmanuel 2001). These results further confirm the gen-

erality of the RKW concepts in understanding the evo-lution of convective systems in a sheared environment.

An important step in judging the success of the pres-ent theory is whether it can be succesfully applied todescribe real squall-line behavior. Within the realm ofidealized simulations, environmental sensitivities can besystematically investigated, and an objective judgmentcan be rendered concerning the types of environmentsmost conducive to system strength and longevity. Com-paring such results to observations, however, can beparticularly difficult, as observations are often quitesparse, and it is also difficult to differentiate the impactof external versus internal forcing mechanisms (e.g.,given sufficient CAPE, a squall line can be long-livedby virtue of continual forcing from a synoptic-scale coldfront, etc., independent of vertical wind shear condi-tions). Cold-pool characteristics are especially difficultto monitor over the lifetime of a system, and such inputis critical for comparison with the above modeling re-sults. One must also be careful to differentiate betweenthe identification of environments that are in some wayoptimal for the maintenance of strong squall lines ascompared to the identification of environments that aremost often observed in association with the strongest,most long-lived systems. For instance, climatologically,environmental shear profiles are generally not restrictedto low levels alone, so it is difficult to gauge the relativeimportance of low-level versus deeper-layer shear fromobservations. Studies such as Evans and Doswell (2001)have begun to address certain important aspects ofsquall-line climatology (e.g., the ability of such systemsto produce strong surface winds) in a systematic manner,but far more work is needed, addressing the other as-pects of squall-line structure discussed above. Directcomparisons between numerical simulations and obser-vations are also made more difficult by ongoing uncer-tainties in the representation of many aspects of modelphysics, such as boundary layer processes, microphys-ics, etc. However, in cases when sufficient observationsof system structure, cold-pool strength, vertical windshear, etc., have been available, we believe that theyhave consistently confirmed the basic shear relation-ships discussed herein (e.g., see discussions in RKW,section 5, and WKR, section 8).

The fundamental issue addressed by RKW, WKR, andthe present study is why environmental vertical windshear promotes stronger, more long-lived squall lines.The fact that a certain amount of shear is beneficial tosquall lines is not a point of controversy. The fact thatlow-level shear is especially beneficial to squall linesdates back to the earliest squall-line simulations studiesof Thorpe et al. (1982), and has been consistently re-produced in all explicit squall-line simulation studiessince. RKW specifically addresses and explains whysuch low-level shear is especially beneficial, but doesnot purport to explain every aspect of squall-line be-havior. System-generated features, such as rear-inflowjets (e.g., Lafore and Moncrieff 1989; Weisman 1992)


or line-end vortices (e.g., Weisman and Davis 1998) arecertainly important components of squall-line structure,but we find that even these features develop as a con-sequence of the simple cold-pool–shear relationshipsespoused herein, which we find represent the most fun-damental internal control on squall-line structure andevolution.

Acknowledgments. We have benefited greatly fromdiscussions and reviews of this manuscript by Drs. Stan-ley B. Trier, David C. Dowell, and Robert J. Trapp andthree anonymous reviewers.

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