Theories on the Optimal Conditions of Long-Lived Squall Lines

Transcription

1 Theories on the Optimal Conditions of Long-Lived Squall Lines References: Thorpe, A. J., M. J. Miller, and M. W. Moncrieff, 1982: Two -dimensional convection in nonconstant shear: A model of midlatitude squall lines. Quart. J. Roy. Meteor. Soc., 108, Rotunno, R., J. B. Klemp, and M. L. Weisman, 1988: A theory for strong long-lived squall lines. J. Atmos. Sci., 45, Lafore, J.-P., and M. 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, Lafore, J.-P., and M. W. Moncrieff, 1990: Reply to Comments on "A numerical investigation of the organization and interaction of the convective and stratiform regions of tropical squall lines". J. Atmos. Sci., 47, Xue, M., 1990: Towards the environmental condition for long-lived squall lines: Vorticity versus momentum. Preprint of the AMS 16th Conference on Severe Local Storms. Amer. Meteor. Soc., Alberta, Canada, Xue, M., 2000: Density current in two-layer shear flows. Quart. J. Roy. Met. Soc., 126, A Schematic Model of a Thunderstorm and Its Density Current Outflow Downdraft Circulation - Density Current in a Broader Sense (Simpson 1997) 1

2 Schematic of a Thunderstorm Outflow (Goff 1976, based on tower measurements) Rotor A Conceptual Model of Quasi-Steady 2D Squall Lines (Thorpe, Miller and Moncrieff 1982) Cold Pool In a Broad Sense 2-D Time-Averaged Flow -- Numerical Simulation 2

3 Theories of Intense / Long-lived Squall Lines Thorpe, Miller and Moncrieff (1982) TMM Theory Rotunno, Klemp and Weisman (1987) RKW Theory Perspective The RKW theory for long-lived squall lines, though widely cited, remains controversial We try to look at more careful look at the theory here 3

4 Key Findings of Thorpe, Miller and Moncrieff TMM82 P0 P-10 P-5 P5 P10 P0 Total Rainfall = 449 P0 is quasi-stationary and produced maximum total precipitation Thorpe, Miller and Moncrieff TMM82 All cases required strong low-level shear to prevent the gust front from propagating rapidly away from the storm; TMM concluded that low-level shear is a desirable and necessary feature for convection maintained by downdraught. 4

5 RKW Theory 5

6 RKW s Vorticity Budget Analysis to Obtain the optimally balanced condition RKW s Vorticity Budget Analysis to Obtain the optimally balanced condition 6

7 RKW Optimal Shear Condition Based On Vorticity Budget Analysis η < 0 w η > 0 u=0 d H u L R u= ur,0 = c ( uη) ( wη) B + = x z x RKW Optimal Shear Condition X R Vorticity Budget Analysis of RKW ( wη ) ddx = 0 η < 0 X L d ( uη ) Ldz = 0 0 H<d w? η > 0 d u ( uη) ( wη) B + = x z x 0 0 d d R R H 0 0 L ( uη) dz ( uη) dz + ( wη) dx R L d 0 0 L ( wη ) dx = ( B B ) dz L 0 R H 0 0 L Bdz g θ / θ H c /2 L 2 d 0 R u ( uη ) Rdz = 2 2 R,0 u= ur,0 = c 7

8 RKW Numerical Experiment of a Spreading Cold Pool Area To be Shown θ' Line-Relative Vectors η, Div (shaded) RKW Density Current Simulation Results Circulation are induced by cold pool propagation, NOT vorticity or shear u=c 8

9 Questions What is the Flow Pattern at the Gust Front? Does the Low-level Inflow have to Contain Shear or Vorticity? Does RKW s Vorticity Balance Argument Adequately Explain the Behavior of Low-level Lifting, Updraft Orientation and Squall Line Intensity? If Not, What Are the Most Important Factors for Strong Low-level Lifting and a Deep Updraft? We will Attempt to Answer These Questions by Using Idealized Density Current Models Numerical Simulations of Realistic Squall Lines Observations First, Theoretical Models of Density Currents 9

10 Inviscid Model of Benjamin (1968, JFM) Density Current Inviscid fluids No vertical shear; Apply Bernoulli s theorem (energy conservation), flow-force balance and mass conservation: h = d/2, c1 = 2 gh ρ/ ρ0 Convention for Environmental Shear β β α α Shear > 0 Shear < 0 10

11 Introduction of Constant Shear (Xu 1992 JAS; Xu, Xue and Droegemeier 1996 JAS) Benjamin s Solution Depth and speed of cold pool increase with increasing positive shear; h > 0.5 H for α > 0; h < 0.5 H for α < 0. Solution verified using timedependent numerical model by Xu, Xue and Droegemeier (1996 JAS). Inviscid Density Current Models in Variable Vertical Shear (Xue, Xu, Droegemeier, 1997 JAS; Xue 1999) Two shear layers allow for more flexibility with inflow configuration, e.g., 11

12 Two Shear-Layer Density Current Model Start from 2-D Boussinesq equations (u, w, θ', ρ') Assumptions Steady State Inviscid, No Surface Friction Far-field flow is horizontal Density constant within and outside cold pool Non-Dimensionalization 2 ρ0 (1) ( xz, ) ( xz, )/ H, ( uw, ) ( uw, )/ U, p' p'/( U ) H is depth of the domain 1/2 U ( gh θ/ θ ) 0 θ θ 1 θ 0 p' P - P 0 and P 0 = gθ 0(H-z). The flow is fully described by seven parameters: { αβ,, d, c, h, d, c}

13 Far-Field Solution We seek far-field (at the left and right boundaries where flow becomes horizontal) steady-state solution by applying four basic principles: (1) Mass Continuity (2) Vorticity Conservation (3) Flow-force Balance and (4) Conservation of Bernoulli Energy along Streamlines. With these constraints, only three of the seven parameters are independent. The Inflow profile is Far Field u Profiles c0 + α d0+ β ( z d0) for d0 < z 1 u ( z) = c0 + α d0 + α ( z d0) for0 z d0. (2) The Outflow profile is given by u c1+ α d1+ β( z h d1) for h+ d1< z 1 () z = c1 + α d1 + α( z h d1) for h z h + d1. (3) 13

14 Bernoulli Theorem Bernoulli Theorem: For a steady-state inviscid flow, the Bernoulli 2 energy ( p' + V /2 ) is conserved along a streamline. p' + 0= 0 + u / B 2 A 2 Flow Force Balance = ( p' u ) dz ( p' u ) dz 14

15 Solution Procedure From the four principles, we obtain four equations They reduce the 7 free parameters to 3 We specify the low-level level and upper-level inflow shear (a and b) ) and the interface height (d 0 ), and seek solutions of the other parameters which describe the cold pool depth, speed etc. The cold pool depth determines, to a large extent, the steepness of the frontal interface and therefore the erectness of the updraft. Sample Far Field Solutions 15

16 Fixed Upper-Level Shear (b = +0.5) and Variable Low-Level Shear Stronger low-level shear --> deeper cold pool Effect is greater for deeper shear h d0 = 0.9 Benjamin s solution (b = +0.5) d0 =.1 Same holds true for frontal speed α Zero Low-Level Shear (a = 0) and Variable Upper-Level Shear Cold pool depth increases with increasing upper-level shear (not considered by RKW) Effect increases with shear layer depth h Benjamin s solution (a = 0) Upper-level shear plays a similar role as the low-level shear β 16

17 Local Structure of the Front The geometric shape of the interface is governed by the dynamic (pressure continuity) condition: D (u 2 + w 2 ) = 2z where D ( ) represents the jump of across the interface. 60 o Summary of Theoretical Results Positive inflow shear, either at low-levels or at upperlevel, supports a deep cold pool, steep frontal interface, and therefore a deep updraft. A deep updraft can be supported even without lowlevel inflow shear The RKW Theory, however, considers the low-level shear essential for deep updraft to form 17

18 Verification of Theory With a Timedependent Numerical Model Is a Steady State Solution Realizable in a Time- Dependent, Fully Nonlinear Model? What Is the Shape of the Frontal Interface and the Orientation of the Updraft? What Effects Do Baroclinically-Generated Frontal Eddies Have on the Flow Structure In the Head Region? We Answer These Questions Using a Time- Dependent Numerical Model, the ARPS. Numerical Studies (XXD96, XXD97, Xue 1999) - Typical Setup of our Experiments Initial cold pool sufficiently large for quasi-steady state to establish; Free from boundary influence; Nose slope set to 60 degrees; Depth set with analytical solution as a guide. Either same as and purposely deviating from the analytical solution; Initial flow obtained from stream-function solved from vorticity equation 18

19 ARPS Model Parameters Two-Dimensional Planar Geometry Boussinesq and other Approximations to Make ARPS Best Match the Idealized Model ~ 800x80 grid points dx = 25 m, dz = 12.5 m Potential Temperature Advected by Monotonic FCT Scheme No Artificial Numerical Mixing for Scalars Turbulent Eddies Explicitly Resolved without SGS Parameterization Results Presented in Non-dimensional Space and Time Zero Upper-Level Shear, Different Low-Level Shear b=0 a= -1 b=0 No Cold Pool Induced Internal Circulation a= +1 Figures Plotted to Scale 19

20 Verification of Theoretical Model Solutions with Numerical Experiments Variation in Low-level Shear Model Specified Theoretical / Model Simulated Expt. α d 0 h 0 h c 0 +αd 0 LS / /0.400 LS / /0.318 LS1A / /0.412 DS / /0.709 SLS / /0.789 SLSA / /0.635 Expt. Model Specified Theoretical / Model Simulated α β d0 h c0 L1U / /1.04 L1U / /0.79 L1UM / /0.45 L1UM / /0.38 L0U / /0.95 L0UM2 0-2 Moderate Low-level Shear, Different Initial Cold Pool Depth Time Averaged Fields Initial Depth = 0.59 b=0 a= +1 Initial Depth = 0.41 b=0 a= +1 20

21 Verification of Theoretical Model Solutions with Numerical Experiments Variation in Low-level Shear Model Specified Theoretical / Model Simulated Expt. α d 0 h 0 h c 0 +αd 0 LS / /0.400 LS / /0.318 LS1A / /0.412 DS / /0.709 SLS / /0.789 SLSA / /0.635 Expt. Model Specified Theoretical / Model Simulated α β d0 h c0 L1U / /1.04 L1U / /0.79 L1UM / /0.45 L1UM / /0.38 L0U / /0.95 L0UM2 0-2 Strong Low-level Shear, No Upper-Level Shear Initial Cold Pool Depth Specified According to Theoretical Solution Initial Cold Pool Depth set as 0.2, Much Shallower Than Theoretical Solution Fields at T = 12 b=0 a= +3 Solution Highly Non-Steady Time Averaged Over 6 Time Units c Time Averaged Streamlines 21

22 Strong Low-level Shear, Different Initial Cold Pool Depth Model Specified Theoretical / Model Simulated Expt. α d 0 h 0 h c 0 +αd 0 LS / /0.400 LS / /0.318 LS1A / /0.412 DS / /0.709 SLS / /0.789 SLSA / /0.635 Expt. Model Specified Theoretical / Model Simulated α β d0 h c0 L1U / /1.04 L1U / /0.79 L1UM / /0.45 L1UM / /0.38 L0U / /0.95 L0UM2 0-2 Zero Low-level Shear with Opposite-Sign Upper- Level Shears β=+2 α=0 β=-2 α=0 Cold pool structure strongly influenced by upperlevel shear too; not considered by RKW 22

23 Model Specified Theoretical / Model Simulated Expt. α d 0 h 0 h c 0 +αd 0 LS1 Verification of Theoretical 0.590/0.538 Model Solutions 0.412/0.400 LS / /0.318 with Numerical Experiments LS1A / /0.412 DS / /0.709 SLS / /0.789 SLSA / /0.635 Variation Variation in in Upper-level Low-level Shear Shear Expt. Model Specified Theoretical / Model Simulated α β d0 h c0 L1U / /1.04 L1U / /0.79 L1UM / /0.45 L1UM / /0.38 L0U / /0.95 L0UM / /0.30 Effect of Cold Pool Internal Circulation 23

24 Effect of Cold Pool Internal Circulation Previous studies (e.g., Xu and Moncrieff 1984) show that, for an inviscid idealized density current in constant shear flow the depth and speed of cold pool are only slightly affected by the internal circulation the solution does NOT depend on the direction of the internal circulation Our numerically simulated density currents behave very differently for + and internal circulations due to the presence of turbulent eddies. Positive Internal Shear γ=1 Negative Internal Shear γ=-1 24

25 T=12 Positive Internal Shear γ=1 Negative Internal Shear γ=-1 No Significant Circulation Induced by Cold Pool Conclusions from Numerical Experiments Theoretical model solutions are verified to a high degree of accuracy by a fully nonlinear, time -dependent numerical model; K-H eddies are explicitly resolved by the numerical model. However, their effect on the density current head is relatively small. The model solutions are robust and are independent of the initial conditions as long as the cold pool air supply is sufficient for the expected depth to be achieved; When insufficient cold air is available, the flow is highly unsteady. However, the flow pattern in a time-averaged sense is still close to the steady solution. This non-steady scenario is more typical of atmospheric squall lines. The overall flow is dictated by the overall vorticity distribution in the domain. Low-level shear is not necessary to establish a deep cold pool (which supports an effective trigger mechanism), as RKW theory suggests. 25

26 Numerical Simulations of Squall Lines in Support of Our Last Argument 2-D Squall Line Simulations of Xue (1989, 1991) Linear Low-level Shear Step Inflow Profile with Zero Vorticty 26

27 Step Profile Cases Time (0-10 Hours) -10m/s Stationary -18m/s Constant Speed Propagation -15m/s -25 m/s Line is Quasi-Stationary 27

28 Time 15 m/s Step Inflow Profile t=5h t=10h x Rainfall Rate from 6 to 10 h Low-level Linear Shear Inflow Cases (0-4 hours) 12m/s 15m/s 20m/s 28m/s 28

29 Conceptual Model of of Xue (1991) c = cloud-relative cold pool speed Line Relative Inflow Profiles RKW Simulation Results θ Line-Relative Vectors η, Div (shaded) Optimal Condition No need for Cold Pool Circulation or Inflow Shear 29

30 Bluestain and Jane (1985) Composite hodograph from 26 cases. Net shear below 2 km is very small. More shear is between 2 7 km. Composite Radar Reflectivity Images of May 08, :00 UTC 02:45 UTC X B5 X DFW 30

31 Observed Pre-Squall Line Wind (U) Profiles, 00 UTC, May 8, 1995 Dallas-Fort Worth Okmulgee, OK Average shear normal to the line is nearly zero in the lowest 2 km in both profiles. January 22-23, 1999 Eastern US Squall Line (60 Tornado Outbreaks in Arkansas 36 hours earlier) Infrared Imagery Showing Squall Line at 12 UTC January 23, ARPS 48 h Forecast at 6 km Resolution Shown are the Composite Reflectivity and Mean Sea-level Pressure. 31

32 Vertical Cross-Section Hardly any cold pool temperature perturbation -> Little buoyancy production of negative vorticity; Convergence at the gust front is due to rotor circulation Dynamic (Bernoulli) effect surpasses thermodynamic effect. ARPS Extracted Pre-Squall Line-Relative U Profile 2km RKW vorticity balance theory would not help 32

33 Summary Point #1 RKW theory considers the circulation at the edge of the cold pool (via baroclinic vorticity generation) to be the essential ingredient in generating a circulation to counter the inflow shear Our high-resolution model simulations show that negative vorticity is confined to a thin vortex sheet that is mostly advected to the rear of the system. The circulation resulting from the vorticity is insignificant at the frontal interface. The cold pool internal circulation has little effect on the frontal slope. The primary role of the cold pool is to decelerate the ground level inflow, rather than generating the so called cold pool circulation. Summary Point #2 RKW optimal condition is based on a vorticity budget analysis that gives a zero flux condition at the top of the control volume, which assumes an updraft profile that may not be achievable RKW assumes that the upper-level air will be calm relative to the gust front, which is not guaranteed by the vorticity balance condition. Our approach tries to solve for the flow around the gust front; Our results also show that upperlevel front-relative flow plays an equally important role in determining the slope of the updraft. 33

34 RKW s optimal balance condition requires that the inflow directly interacting with the cold pool contain positive vorticity that matches the negative vorticity generated by the gust front; Summary Point #3 Our theoretical model, confirmed by numerical simulations of both density currents and squall lines, shows that, despite the presence of a cold pool, deep, long-lasting and quasi-steady updrafts can be established without low-level inflow shear. Conclusions Low-level inflow shear is NOT necessary for establishing deep, steady updrafts Baroclinically generated cold pool circulation does not appear to have significant effect on the structure of density current head Rather, the shear between the ground level and the steering level is a more important factor in determining the propagation of cold pool relative the cloud system above The updraft orientation is a function of vorticity distribution throughout the entire domain, and a global solution should be obtained by solving the vorticity equation with proper boundary conditions. To determine the behavior of the updraft branch of inflow over the cold pool, we need to know the vorticity distribution in the entire domain and the boundary conditions. Vorticity in an air parcel alone cannot tell us its trajectory. 34

35 Conclusions In general, a cold pool that propagates at the speed of, or slightly faster than, the steering level wind (or the propagation speed of a cloud) creates an optimal condition for intense, long-lasting squall lines. The role of the low-level system relative inflow is to prevent the cold pool from propagating away from the overhead cloud. The surface system-relative wind speed, rather than the shear, is most important. Our optimal condition based on front propagation speed and surface and steering level winds makes few assumptions and is more generally valid. 35