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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 non-constant shear: A model of midlatitude squall lines. Quart. J. Roy. Meteor. Soc., 108, 739-762.

• Rotunno, R., J. B. Klemp, and M. L. Weisman, 1988: A theory for strong long-lived squall lines. J. Atmos. Sci., 45, 463-485.

• 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, 52-1544.

• 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, 1034-1035.

• 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, 24-29.

• Xue, M., 2000: Density current in two-layer shear flows. Quart. J. Roy. Met. Soc., 126, 1301-1320.

A Schematic Model of a Thunderstorm and Its Density Current Outflow

Downdraft Circulation- Density Current in a Broader Sense

(Simpson 1997)

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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

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Key Findings of Thorpe, Miller and Moncrieff 1982 - TMM82

P0 is quasi-stationary and produced maximum total precipitation

P-10 P-5 P5 P10

P0

P0Total Rainfall

= 449

Thorpe, Miller and Moncrieff 1982 - 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 featurefor convection maintained by downdraught.

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RKW Theory

RKW’s Vorticity Budget Analysis to Obtain the ‘optimally’ balanced condition

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RKW’s Vorticity Budget Analysis to Obtain the ‘optimally’ balanced condition

In the above, c is defined by

which is exactly the density current propagation speed we derived earlier! Therefore the optimal condition obtained based on RKW’s vorticity budget analysis says that the shear

magnitude in the low-level inflow should be equal to the cold pool propagation speed.

20 0 0

0

2 ( ) 2 2

2

HLc B dz g H g H

c g H

θ ρθ ρ

ρρ

−∆ ∆= − = =

∆=

∫

RKW Optimal Shear Condition Based On Vorticity Budget Analysis

,0Ru u c∆ = − =

∆u

L R

H

d

w0η >0η <

u=0

( ) ( )u w Bx z xη η∂ ∂ ∂

+ = −∂ ∂ ∂

6

u

L R

H<d

d

RKW Optimal Shear Condition

Vorticity Budget Analysis of RKW

2,0

0

( )2

dR

R

uu dzη = −∫

0

( ) 0d

Lu dzη =∫

20

0

/ / 2H

LB dz g H cθ θ≈ − ∆ ≡ −∫

( ) ( )u w Bx z xη η∂ ∂ ∂

+ = −∂ ∂ ∂

0 0

,0Ru u c∆ = − =

0 0

00

( ) ( ) ( )

( ) ( )

d d R

R L dL

R H

L RL

u dz u dz w dx

w dx B B dz

η η η

η

− +

− = −

∫ ∫ ∫

∫ ∫0

( ) 0R

L

X

dX

w dxη =∫ w0η >0η <

?

RKW Numerical Experiment of a Spreading Cold Pool

Area To be Shown

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θ' η, Div (shaded)Line-Relative Vectors

RKW Density CurrentSimulation Results

∆u=cCirculation are induced by cold pool propagation, NOT vorticity or shear

Questions

• Does the Low-level Inflow have to Contain Shear or Vorticity?

• Can long-lived squall lines be supported even without shear in the lowest few kms of the troposphere?

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First, Theoretical Models of Density Currents

Inviscid Steady-state Density Current Models in Variable Vertical Shear(Xue, Xu, Droegemeier, 1997 JAS; Xue 2000 QJ)

Two shear layers allow for more flexibility with inflow configuration, e.g.,

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Summary of Theoretical Model Results

• Positive inflow shear, either at low-levels or at upper-level, supports a deep cold pool, steep frontal interface, and therefore a deep updraft.

• A deep updraft can be supported even without low-level inflow shear

• The RKW Theory, however, considers the low-level shear essential for deep updraft to form

Results from a Time-dependent Numerical Model

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Zero Upper-Level Shear, Different Low-Level Shear

α= -1

α= +1

β=0

β=0

Figures Plotted to Scale

No Cold Pool Induced Internal Circulation

Zero Low-level Shear with Opposite-Sign Upper-Level Shears

Cold pool structure strongly influenced by upper-level shear too; not considered by RKW

α=0

α=0

β=+2

β=-2

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Conclusions from Numerical Experiments

• 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, contrary to what RKW theory suggests.

Numerical Simulations of Squall Linesin Support of Our Last Argument

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2-D Squall Line Simulations of Xue (1989, 1991)

Linear Low-level Shear Step Inflow Profile with Zero Vorticty

-10m/s -15m/s

-18m/s -25 m/s

Tim

e (0

-10

Hou

rs)

Stationary

Constant Speed

Propagation

Step Profile Cases

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Line is Quasi-Stationary

Low-level Linear Shear Inflow

Cases(0-4 hours)

12m/s

20m/s 28m/s

15m/s

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Conceptual Model of of Xue (1991)

c = cloud-relative cold pool speed

Lin

e R

elat

ive

Inflo

w P

rofil

es

θ’ η, Div (shaded)Line-Relative Vectors

RKW Simulation Results

Optimal ConditionNo need for

‘Cold Pool Circulation’or ‘Inflow Shear’

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Conclusions of Xue et al

• 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.

Conclusions Xue et al – continued…

• 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.