Presentation on theme: "RKW Theory Applied To Forecasting Severe Squall Lines and Mesoscale Convective Systems (MCSs) Mike Coniglio National Severe Storms Laboratory March 30,"— Presentation transcript:
RKW Theory Applied To Forecasting Severe Squall Lines and Mesoscale Convective Systems (MCSs) Mike Coniglio National Severe Storms Laboratory March 30, 2015 Photo by Douglas Berry
PART I: Brief overview of MCS cold pool/shear interactions PART II: Applications of cold pool/shear theories to MCS forecasting PART III: Other considerations for real MCS events (very brief) Narrative modeled after Bryan et al. (2012) SLS and Coniglio et al. (2012) OUTLINE
-We’ll focus on theories for the interaction of convective cells with the ambient vertical wind shear -Note: Use term “shear” to denote units of m s -1 Focus on systems w/nearly contiguous line/arc of deep convection > 100 km that lasts at least 5 continuous hours that produce severe winds Squall lines and “MCSs” for this lecture Parker and Johnson (2000) Also bow echoes Leading stratiform (LS) Parallel stratiform (PS) Trailing stratiform (TS)
Squall lines persist partly by “the continuous generation of new thunderstorms as a result of convergence- divergence patterns produced by the vertical transfer of horizontal momentum in pre-existent thunderstorms.” Newton (1950) Ignored cold pool effects
A squall line is a collection of long-lived cells that deposit their rain upshear allowing the cells and the line to continue indefinitely Ludlum (1963), Newton (1966) Observational Model But early 2D numerical models couldn’t replicate a long-lived system in deep shear
Some 2D simulations allow for long-lived systems if the wind shear was restricted to low-levels Physical interpretation: Low-level shear keeps the cold pool from propagating rapidly away from the storm 2D supercell Hane (1973), Thorpe et al. (1982) Problem: Many squall lines don’t contain steady convective cells, either 2D or 3D
Low-level shear contains vorticity that opposes the vorticity generated along the gust front - maximizes likelihood of triggering new cells…..let’s quantify this… 2D supercell (Thorpe et al. 1982) 2D multicell Rotunno et al. (1988) JAS - “RKW”
Quantify flow characteristics of cold pool in shear RKW described the flow using 2D horizontal vorticity ( ) equation RKW quantitative criterion -Steady, Boussinesq flow Neutral ( const) -No friction or mixing u R,0 u R,d u L,0 u L,d u L,H z x - No motion inside cold pool Const in cold pool - Follow cold pool (really only along interface) TO THE BOARD!!!
u R,0 u R,d u L,0 u L,d u L,H z x (really only along interface) RKW: “Net buoyant generation of negative vorticity in the volume is just balanced by the import of positive vorticity at x = R.”
u = 20 m s -1 (over lowest 2.5 km) u = 30 m s -1 (over lowest 2.5 km) strongest lift magnitude of cold pool vorticity ≈ ambient vorticity, or c ≈ Δu magnitude of cold pool vorticity < ambient vorticity, or c < Δu all cold pool vorticity, no ambient vorticity, or c ∞> Δu u = 0 m s -1 Flow structure reproduced in 2D numerical models in RKW: x x z z z Contours are shading is convergence shading is neg pert pot temp
“weak” shear: u = 10 m s -1 (over lowest 2.5 km), from RKW strong updrafts in early evolution …and the system overall weakens with new cells further and futher behind the gust front 240 min270 min 300 min330 min Θ e shaded x x z z …eventually weaken with time as the cold pool “overwhelms” the shear… Thought: When the low-level shear is weak, the system sows the seeds to it’s own demise through the cold pool
u (m s -1 ) over lowest 2.5 km Time-dependent relationship seen in early 2D and 3D numerical models: * - To be revisited systems with “small” u weaken steadily with time from RKW Cold pools are “inimical to nonsupercellular updrafts”, meaning cold pool air prevents parcels from realizing their full potential…it’s hard to maintain optimal state Time (min) W max (m s -1 ) *
u R,0 u R,d u L,0 u L,d u L,H z x ?
Does d (height of the shear layer) matter? For severe weather applications, it certainly does. u calculated over only the lowest 2.5 km is often quite different than u computed, say, over 5 km If we use the RKW and follow-on simulations as a guide for expected real MCS behavior, this can mean the difference between non-severe and severe event, or at least a different convective organization
Does d (height of the shear layer) matter? RKW: “Since cold pools are located at low levels, the shear can only promote lifting when restricted to low levels…” and “…it is the actual shear that makes contact with the cold pool that is influential, not simply the velocity difference between the subcloud and cloud-bearing layer” from Weisman et al. (1988) overall convective response for the deep- shear experiments is significantly weaker u (m s -1 )
Summary of understanding at the time: Low-level shear counters the negative effects of a cold pool on triggering new cells “Optimal” case: When positive vorticity in the inflow just balances negative vorticity generated by cold pool (upright updrafts), or C ≈ u with u confined to the ~ depth of the cold pool If shear is too small, system leans upshear quickly and weakens steadily with time as cold pool depletes CAPE Not just for squall lines: Strong systems w/bow echoes need strong low- level shear (> 20 m s -1 /2.5 or 5 km) and are favored if the shear is confined to low levels (Weisman 1993) …return to later “If an appropriate balance between the low-level shear and the cold pool is ‘only part of the problem’, we believe it is a vital part, and is in fact, a necessary (clearly not a sufficient) requirement for sustaining most strong squall lines.” (Rotunno et al JAS, comment).
That’s the basic theory for maximizing lift along a cold pool, what about it’s application to forecasting? Behavior of squall lines and MCSs: 1.Longevity 2.Strength 3.Structure
SPC: Squall lines and MCSs are often strong and long-lived despite significantly sub-optimal conditions: Fovell and Ogura (1989) 2D simulations replicated qualitative behavior of squall lines with varying low-level shear in RKW, but…. “…none of the model storms convincingly exhibited a decaying phase following its mature phase…” u (m s -1 ) Over 8 h 3D simulations
Is RKW really a theory for the longevity of squall lines/MCS? RKW: If shear is too small, system leans upshear quickly and weakens steadily with time as cold pool depletes CAPE Fovell and Ogura (1989): No! The above time-dependent behavior may be mostly a symptom of a small domain size! (numerical error related to open boundary conditions)
Supported in Coniglio and Stensrud (2001) in 3D simulation with horizontally inhomogeneous environment derived from composite analyses “optimal” “…intense convective updrafts persist in the mid- and upper levels directly above the gust front for many hours, despite significantly less than optimal conditions”….. Sub-optimal shear Time (h) “updrafts are not being depleted of their CAPE by the cold pool…weakening of the system is controlled more by the CAPE.”
Confirmed in Weisman and Rotunno (2004) and Bryan et al. (2006) RKW90: “… low-level shear is, in fact, a necessary (clearly not a sufficient) requirement for sustaining most strong squall lines”, BUT…. WR04: “Squall-line longevity, without reference to system strength and structure, is not found to be as sensitive to shear as in the past studies.” Bryan et al. (2006): “…system longevity can no longer be considered a part of [the RKW] argument.”
Coniglio et al. (2007 & 2010): In observed soundings, for MCSs with strong leading convective line, low-level shear doesn’t discriminate nearly as well as deeper-shear for MCS maintenance: And instability plays an important role of course… “MCS Maintenance Parameter” on SPC Mesoanalysis has large contribution from these two variables (Coniglio et al. 2007)
Basic theory for maximizing lift along a cold pool, what about it’s application to forecasting? Behavior of squall lines and MCSs: 1.Longevity 2.Strength 3.Structure
Weisman et al. (1993) In idealized simulations, bow echoes only generated for CAPE ≥ 2000 m 2 s -2 and u ≥ 20 m s -1, but are favored if u is confined to the lowest 2.5 km AGL Weak cells “W” Bow Echo “B” Split cells “S” CAPE (m 2 s -2 ) u (m s -1 ) d = 2.5 km d = 5 km
SPC forecasters in 1990s: Severe, long-lived MCSs (derechos) often occur with both low-level and deeper shear that is weaker than what’s required for strong, long-lived systems in 3D simulations: Evans and Doswell (2001): 113 proximity soundings 75% have 0-3 km shear < 15 m s -1 -Drops to 12 m s -1 for weakly forced events 75% have 0-6 km shear < 20 m s -1 Counter from Weisman (1993): In simulations, severe near- surface winds are produced for Δu as low as 10 m s -1 But, “the convective systems in such cases…evolve more quickly than the stronger-shear cases, as the rear-inflow jet descends to the surface well behind the leading edge of the system, thereby promoting shallower lifting along the gust front.” i.e. Need strong low-level shear for strong, long-lived bow echoes in simulations
Another Counter from Weisman (1992): The RKW optimal condition doesn’t include the effects of the rear-inflow jet, a feature of many squall lines and really all bow echoes
u R,0 u R,d u L,0 u L,d u L,H z x Weisman (1992,1993): Case with elevated rear-inflow jet (u L,H = u L,d > 0) RIJ -RKW: “Net buoyant generation of negative vorticity in the volume is just balanced by the import of positive vorticity at x = R and the positive vorticity in the cold pool.”
-RIJ forms between sloping updraft and cold pool due to buoyancy gradients (and maintained by bookend vortices) -Vorticity associated with elevated RIJs help to “prop-up” the updrafts to promote vigorous convection, but… -only occurs in simulations if shear is strong, and subsequently… -RIJs need to be > 10 m s -1 (relative to cold pool motion) to have significant impact, thus “the RKW theory alone (without the addition of the rear-inflow current) may still be sufficient to characterize the evolution of most observed convective systems.” Weisman (1993) -Really, a good thing for applying RKW theory because it’s hard to quantify RIJ strength
-Another problem with application is calculation of -θ’ at surface may be helpful if decreases linearly w/height, but really need a sounding inside the cold pool and in ambient environment for accurate estimate -But the behavior of simulated squall lines/MCSs for different shear values/depth may still be helpful, especially if cold pools in simulations are in the ball park (are they?)
Klemp-Wilhelmson (KW) Numerical Model (Klemp and Wilhelmson 1978) Simulations of squall-lines using KW model typically have cold pools ~2 km deep and max ~ 8 K ( Engerer et al ): Bryan et al. (2006): But KW model cold pools are actually artificially too strong and too deep because of too much vertical diffusion at the lower boundary – acts as an artificial heat sink. In idealized KW model with WK82 profile, cold pools are really ~1 – 2 km deep with accurate diffusion (Bryan et al. 2006).
RKW: “Since cold pools are located at low levels, the shear can only promote lifting when restricted to low levels…” and “…it is the actual shear that makes contact with the cold pool that is influential, not simply the velocity difference between the subcloud and cloud-bearing layer” Does this mean u should be evaluated in lowest 1 – 2 km? This would make the discrepancy in behavior of observed and simulated squall lines for different values of u even greater! WR04: The above statement (in red) is “technically incorrect”!! “The phrase should have been ‘in close proximity to’…” This is largely because of an “action at a distance” argument that was not emphasized in original RKW paper- flow can be influenced away from the vorticity source (elliptic equation). Weisman and Rotunno (2004)
So, what does “in close proximity to” mean? Important for applications… Not quantified in WR04, but they use u over the lowest 5 km to evaluate RKW theory and confirm the optimal condition (c ≈Δu) (as does Bryan et al. 2006), despite model cold pools that are ~ km deep, or over 2 times the cold pool depth now! So evaluate u over 2 times the cold pool depth…..begs the question how strong and deep are real MCS cold pools? Engerer et al. (2008) Oklahoma MCSs RKW models C (m s -1 ) Cold pool depth (km) Bryan et al. (2005) BAMEX MCSs RKW models
So, if we should use u over 2 times the cold pool depth to evaluate RKW theory, and if many MCS cold pools are really ~3 – 4 km deep, does this mean we should look at u over 6 – 8 km!? If so, this interpretation is more consistent with observations (shown earlier, and…) 5-10 km shear (m s - 1 ) strength of synoptic forcing: Coniglio et al. (2004) Environments of derechos Cohen et al. (2007) Environments of weak, strong, and derecho MCSs line-perpendicular shear (m s -1 ) weak strong derecho shear layer (km)
0000 UTC 28 MAY 2001 OUN CAPE ~ 3800 J/kg km shear ~ 11 m/s 0-5-km shear ~20 m/s 5-10-km shear ~30 m/s 0230 UTC 28 MAY 2001 X May 2001 derecho Photo by Douglas Berry
4 July 2003 Derecho SLA WDL SLA km shear ~ 8 m/s 0-5-km shear ~18 m/s 5-10-km shear ~20 m/s km shear ~ 4 m/s 0-5-km shear ~16 m/s 5-10-km shear ~12 m/s
This interpretation (look at u over deep layers) is also consistent with other factors that can control squall line/MCS strength, longevity, and structure Lafore and Moncrieff (1989): “…the wind shear in the entire troposphere should be considered in the interaction of the cold pool and the convective cells…the organization of the vorticity fields is predominantly on the scale of the entire system…” Fovell and Dailey (1995): “…positive shear throughout the atmosphere below cloud top is potentially influential on updraft airflow intensity and orientation…” Fovell and Tan (1998), Weisman and Rotunno (2004): Convective cells penetrate into mid-upper levels and induce buoyancy gradients that interact with shear there Shapiro (1992): Sheared flow just needs a barrier to overturn (no negative buoyancy acting at a distance needed) Parker and Johnson (2004), Coniglio et al. (2006): Updrafts in shear induce perturbation low- pressure on downshear side…act to “prop-up” cells
Using u over deep layers in forecasting may be better than examining only the shear over low-levels because it’s an integrated measure of many factors over the trop that can influence squall line/MCSs (Coniglio et al. 2007) Having shear confined to low-levels (or depth of cold pool) may very well still be “optimal”, but “may actually be quite rare in nature” (Stensrud et al. 2005, Bryan et al. 2014) Avg./max surface wind (m s -1 ) From Weisman and Rotunno (2004)
Other factors and considerations
Many strong squall-lines and MCSs occur at night in presence of nocturnal LLJ Johns (1993) Fritsch and Forbes (2001) Low-level shear vorticity induced by LLJ is negative. If positive vorticity in shear is helpful in RKW sense, it’ll be above the LLJ level (i.e. effective shear layer) But LLJ forcing may be just as important as cold pool/shear interactions (French and Parker 2010, Coniglio et al. 2012)
Very strong and deep LLJ Strong squall line/bow echo developed and matured despite shear pointed toward the line in the lowest 2 – 3 km From Coniglio et al. (2012) May 8, 2009 LMN profiler m s -1 at 1 – 3 km AGL
BUT, I’ve focused on severe MCSs (subject of this class)….what about for other classes of squall lines/MCSs? RKW may be much more of a factor in cases in daytime boundary layers away from strong external forcing and shallow cold pools, e.g. Cold-frontal rainbands (Carbone 1982) Tropical squall lines (Roux 1988) Squall lines over eastern U.S. or w/pre-frontal troughs (e.g. Lombardo and Colle 2010) –Weaker cold pools & shallower shear profiles –but then often need to worry about terrain effects (e.g. Frame and Markowski 2006, Letkewicz and Parker 2010, 2011)
“If an extremely short description of RKW theory is needed, then it is probably best to use the phrase ‘A Theory for Squall Line Structure’” (Bryan et al. 2012) This qualitative behavior of C versus u still holds for describing overall tilt, but: u should be depicted over at least twice the depth for application 2.Use u in effective inflow (exclude stable layers) 3.Not a control on longevity alone, and strong to severe MCSs can occur for a wide range of C/ u (0.5 to 3 as suggested in Bryan et al. 2012)
Many other factors only touched on (or ignored) here: Shear often varies across the convective line as MCSs often occur on equatorward edge of mid-upper-level jet streaks Spatial distribution of high lapse rate/low CIN/high PW axes Orientation of shear relative to mean wind and instability axis (Corfidi lecture) LLJ forcing Forcing from favorable juxtaposition of upper-level jet streak prevents long-lived MCS: A failure mode (still pondering) –right entrance for mesoscale lifting –subsidence with right exit to suppress release of instability prior to MCS cold pool arriving Upper-level Inertial Instability can favor fast upscale growth and MCS modes (Coniglio et al. 2010) Forcing from gravity waves and bores (Parker 2008, Trier et al. 2011) Still others…
Questions NWC Room 2237 Mike Coniglio National Severe Storms Laboratory March 30, 2015 Photo by Douglas Berry
References (page 1 of 5) Bryan, G. H., D. Ahijevych, C. Davis, S. Trier, and M. Weisman, 2005: Observations of cold pool properties in mesoscale convective systems during BAMEX. Preprints, 11th Conf. on Mesoscale Processes, Albuquerque, NM, Amer. Meteor. Soc., JP5J.12. [Available online at George H. Bryan, Jason C. Knievel, and Matthew D. Parker, 2006: A Multimodel Assessment of RKW Theory’s Relevance to Squall-Line Characteristics. Mon. Wea. Rev., 134, 2772–2792. George H. Bryan, Richard Rotunno, and Morris L. Weisman, 2012: What is RKW Theory?. Preprints, 26 th Conf. on Severe Local Storms, Nashville, TN, Amer. Meteor. Soc., 4B6 [Available online at https://ams.confex.com/ams/26SLS/webprogram/Paper html.]. https://ams.confex.com/ams/26SLS/webprogram/Paper html George H. Bryan and Richard Rotunno, 2014: The Optimal State for Gravity Currents in Shear. J. Atmos. Sci., 71, 448– 468. Ariel E. Cohen, Michael C. Coniglio, Stephen F. Corfidi, and Sarah J. Corfidi, 2007: Discrimination of Mesoscale Convective System Environments Using Sounding Observations. Wea. Forecasting, 22, 1045–1062. Michael C. Coniglio and David J. Stensrud, 2001: Simulation of a Progressive Derecho Using Composite Initial Conditions. Mon. Wea. Rev., 129, 1593–1616. Michael C. Coniglio, David J. Stensrud, and Michael B. Richman, 2004: An Observational Study of Derecho-Producing Convective Systems. Wea. Forecasting, 19, 320–337. Michael C. Coniglio, David J. Stensrud, and Louis J. Wicker, 2006: Effects of Upper-Level Shear on the Structure and Maintenance of Strong Quasi-Linear Mesoscale Convective Systems. J. Atmos. Sci., 63, 1231–1252.
References (page 2 of 5) Michael C. Coniglio, Harold E. Brooks, Steven J. Weiss, and Stephen F. Corfidi, 2007: Forecasting the Maintenance of Quasi-Linear Mesoscale Convective Systems. Wea. Forecasting, 22, 556–570. Michael C. Coniglio, Jason Y. Hwang, and David J. Stensrud, 2010: Environmental Factors in the Upscale Growth and Longevity of MCSs Derived from Rapid Update Cycle Analyses. Mon. Wea. Rev., 138, 3514–3539. Corrigendum, Mon. Wea. Rev., 139, 2686–2688. Michael C. Coniglio, Stephen F. Corfidi, and John S. Kain, 2012: Views on Applying RKW Theory: An Illustration Using the 8 May 2009 Derecho-Producing Convective System. Mon. Wea. Rev., 140, 1023–1043. Nicholas A. Engerer, David J. Stensrud, and Michael C. Coniglio, 2008: Surface Characteristics of Observed Cold Pools. Mon. Wea. Rev., 136, 4839–4849. Jeffry S. Evans and Charles A. Doswell III, 2001: Examination of Derecho Environments Using Proximity Soundings. Wea. Forecasting, 16, 329–342. Robert G. Fovell and Yoshi Ogura, 1989: Effect of Vertical Wind Shear on Numerically Simulated Multicell Storm Structure. J. Atmos. Sci., 46, 3144–3176. Robert G. Fovell and Peter S. Dailey, 1995: The Temporal Behavior of Numerically Simulated Multicell-Type Storms. Part I. Modes of Behavior. J. Atmos. Sci., 52, 2073–2095. Robert G. Fovell and Pei-Hua Tan, 1998: The Temporal Behavior of Numerically Simulated Multicell-Type Storms. Part II: The Convective Cell Life Cycle and Cell Regeneration. Mon. Wea. Rev., 126, 551–577.
References (page 3 of 5) Jeffrey Frame and Paul Markowski, 2006: The Interaction of Simulated Squall Lines with Idealized Mountain Ridges. Mon. Wea. Rev., 134, 1919–1941. Adam J. French and Matthew D. Parker, 2010: The Response of Simulated Nocturnal Convective Systems to a Developing Low-Level Jet. J. Atmos. Sci., 67, 3384–3408. J. M. Fritsch and and G. S. Forbes, 2001: Mesoscale Convective Systems. Meteorological Monographs, 28, 323–358. Carl E. Hane, 1973: The Squall Line Thunderstorm: Numerical Experimentation. J. Atmos. Sci., 30, 1672–1690. Robert H. Johns, 1993: Meteorological Conditions Associated with Bow Echo Development in Convective Storms. Wea. Forecasting, 8, 294–299. Joseph B. Klemp and Robert B. Wilhelmson, 1978: The Simulation of Three-Dimensional Convective Storm Dynamics. J. Atmos. Sci., 35, 1070–1096. Jean-Philippe Lafore and Mitchell W. Moncrieff, 1989: A Numerical Investigation of the Organization and Interaction of the Convective and Stratiform Regions of Tropical Squall Lines. J. Atmos. Sci., 46, 521–544. Casey E. Letkewicz and Matthew D. Parker, 2010: Forecasting the Maintenance of Mesoscale Convective Systems Crossing the Appalachian Mountains. Wea. Forecasting, 25, 1179–1195. Casey E. Letkewicz and Matthew D. Parker, 2011: Impact of Environmental Variations on Simulated Squall Lines Interacting with Terrain. Mon. Wea. Rev., 139, 3163–3183.
References (page 4 of 5) Kelly A. Lombardo and Brian A. Colle, 2010: The Spatial and Temporal Distribution of Organized Convective Structures over the Northeast and Their Ambient Conditions. Mon. Wea. Rev., 138, 4456–4474. F.H. Ludlum, 1963: Severe Local Storms: A Review. Meteor. Monogr., 5, Amer. Meteor. Soc., C. W. Newton, 1950: STRUCTURE AND MECHANISM OF THE PREFRONTAL SQUALL LINE. J. Meteor., 7, 210– 222. C.W. Newton, 1966: Circulations in Large, Sheared Cumulonimbus. Tellus, 18, Matthew D. Parker and Richard H. Johnson, 2000: Organizational Modes of Midlatitude Mesoscale Convective Systems. Mon. Wea. Rev., 128, 3413–3436. Matthew D. Parker and Richard H. Johnson, 2004: Simulated Convective Lines with Leading Precipitation. Part I: Governing Dynamics. J. Atmos. Sci., 61, 1637–1655. Matthew D. Parker, 2008: Response of Simulated Squall Lines to Low-Level Cooling. J. Atmos. Sci., 65, 1323–1341. Richard Rotunno, Joseph B. Klemp, and Morris L. Weisman, 1988: A Theory for Strong, Long-Lived Squall Lines. J. Atmos. Sci., 45, 463–485. Richard Rotunno, Joseph B. Klemp, and Morris L. Weisman, 1990: Comments on “A Numerical Investigation of the Organization and Interaction of the Convective and Stratiform Regions of Tropical Squall Lines”. J. Atmos. Sci., 47, 1031–1033.
References (page 5 of 5) Alan Shapiro, 1992: A Hydrodynamical Model of Shear Flow over Semi-Infinite Barriers with Application to Density Currents. J. Atmos. Sci., 49, 2293–2305. David J. Stensrud, Michael C. Coniglio, Robert P. Davies-Jones, and Jeffry S. Evans, 2005: Comments on “‘A Theory for Strong Long-Lived Squall Lines’ Revisited”. J. Atmos. Sci., 62, 2989–2996. Thorpe, A. J., Miller, M. J. and Moncrieff, M. W., 1982: Two-dimensional convection in non-constant shear: A model of mid-latitude squall lines. Q.J.R. Meteorol. Soc., 108: 739–762. Stanley B. Trier, John H. Marsham, Christopher A. Davis, and David A. Ahijevych, 2011: Numerical Simulations of the Postsunrise Reorganization of a Nocturnal Mesoscale Convective System during 13 June IHOP_2002. J. Atmos. Sci., 68, 2988–3011. M. L. Weisman and J. B. Klemp, 1982: The Dependence of Numerically Simulated Convective Storms on Vertical Wind Shear and Buoyancy. Mon. Wea. Rev., 110, 504–520. Morris L. Weisman, Joseph B. Klemp, and Richard Rotunno, 1988: Structure and Evolution of Numerically Simulated Squall Lines. J. Atmos. Sci., 45, 1990–2013. Morris L. Weisman, 1992: The Role of Convectively Generated Rear-Inflow Jets in the Evolution of Long-Lived Mesoconvective Systems. J. Atmos. Sci., 49, 1826–1847. Morris L. Weisman, 1993: The Genesis of Severe, Long-Lived Bow Echoes. J. Atmos. Sci., 50, 645–670. Morris L. Weisman and Richard Rotunno, 2004: “A Theory for Strong Long-Lived Squall Lines” Revisited. J. Atmos. Sci., 61, 361–382.