Theories on the Optimal Conditions of Long-Lived Squall Linespascal.sca.uqam.ca/~eva/SCA4662/NouveauxDocuments/METR443/Chapter3... · RKW Numerical Experiment of a Spreading Cold

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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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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)

2-D Time-Averaged Flow -- Numerical Simulation

Cold Pool In a Broad Sense

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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

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

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

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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∂ ∂ ∂

+ = −∂ ∂ ∂

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 <

?

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RKW Numerical Experiment of a Spreading Cold Pool

Area To be Shown

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

RKW Density CurrentSimulation Results

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

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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

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Inviscid Model of Benjamin (1968, JFM)

• Inviscid fluids• No vertical shear;• Apply Bernoulli’s theorem (energy conservation), flow-force

balance and mass conservation:h = d/2, 1 02 /c gh ρ ρ= ∆

Density Current

Convention for Environmental Shear

Shear > 0 Shear < 0

α

β

α

β

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Introduction of Constant Shear(Xu 1992 JAS; Xu, Xue and Droegemeier 1996 JAS)

• 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 time-

dependent numerical model by Xu, Xue and Droegemeier (1996 JAS).

Benjamin’sSolution

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.,

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Two Shear-Layer Density Current Model

Start from 2-D Boussinesq equations(u, w, θ', ρ')

AssumptionsSteady StateInviscid, No Surface FrictionFar-field flow is horizontalDensity constant within and outside cold pool

20( , ) ( , ) / , ( , ) ( , ) / , ' '/( )x z x z H u w u w U p p Uρ← ← ← (1)

H is depth of the domain

1 / 20( / )U gH θ θ≡ ∆

∆θ ≡ θ1 − θ0

p' ≡ P - P0 and P0 = gθ0(H-z). The flow is fully described by seven parameters: 0 0 1 1{ , , , , , , }d c h d cα β .

Non-Dimensionalization

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We seek far-field (at the leftand right boundaries whereflow becomes horizontal)steady-state solution byapplying four basicprinciples:

(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 sevenparameters are independent.

Far-Field Solution

The Inflow profile is

0 0 0 0

0 0 0 0

( ) for 1( )

( ) f o r 0 .

c d z d d zu z

c d z d z d

α βα α∞

− + + − < ≤= − + + − ≤ ≤ (2)

The Outflow profile is given by

1 1 1 1

1 1 1 1

( ) for 1( )

( ) for .

c d z h d h d zu z

c d z h d h z h d

α βα α−∞

− + + − − + < ≤= − + + − − ≤ ≤ + (3)

Far Field u Profiles

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Bernoulli Theorem

Bernoulli Theorem: For a steady-state inviscid flow, the Bernoulli energy ( ) is conserved along a streamline.2' / 2p V+

2 2' 0 0 /ABp u+ = +

Flow Force Balance

1 12 20 0( ' ) ( ' )p u dz p u dz−∞ −∞ ∞ ∞+ = +∫ ∫

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Solution Procedure

•• From the four principles, we obtain four equationsFrom the four principles, we obtain four equations

•• They reduce the 7 free parameters to 3They reduce the 7 free parameters to 3

•• We specify the lowWe specify the low--level and upperlevel and upper--level inflow level inflow shear (shear (αα and and ββ ) and the interface height (d) and the interface height (d00), and ), and seek solutions of the other parameters which seek solutions of the other parameters which describe the cold pool depth, speed etc.describe the cold pool depth, speed etc.

•• The cold pool depth determines, to a large extent, The cold pool depth determines, to a large extent, the steepness of the frontal interface the steepness of the frontal interface and therefore and therefore the erectness of the updraftthe erectness of the updraft..

Sample Far Field Solutions

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• Stronger low-level shear --> deeper cold pool

• Effect is greater for deeper shear

• Same holds true for frontal speed

d0 = 0.9

Benjamin’ssolution

Fixed Upper-Level Shear (β = +0.5) and Variable Low-Level Shear

h

α

d0 = .1

(β = +0.5)

• Cold pool depth increases with increasing upper-level shear (notconsidered by RKW)

• Effect increases with shear layerdepth

• Upper-level shear plays a similar role as the low-level shear

Benjamin’ssolution

Zero Low-Level Shear (α = 0) and Variable Upper-Level Shear

h

β

(α = 0)

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Local Structure of the Front

The geometric shape of the interface is governed by the dynamic (pressure continuity) condition:

∆ (u2 + w2) = 2z

where ∆ ( ) represents the jump of across the interface.

60o

Summary of Theoretical 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

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Verification of Theory With a Time-dependent 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 TimeWe Answer These Questions Using a Time--Dependent Numerical Model, the ARPS.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

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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

α= -1

α= +1

β=0

β=0

Figures Plotted to Scale

No Cold Pool Induced Internal Circulation

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Verification of Theoretical Model Solutions with Numerical Experiments

Model Specified Theoretical / Model SimulatedExpt. α d0 h0 h c0+αd0

LS1 1.0 0.2 0.590 0.590/0.538 0.412/0.400LS2 -1.0 0.2 0.410 0.410/0.325 0.318/0.318LS1A 1.0 0.2 0.410 0.590/0.531 0.412/0.412DS 1.0 0.5 0.683 0.683/0.638 0.696/0.709SLS 3.0 0.2 0.767 0.767/0.763 0.777/0.789SLSA 3.0 0.2 0.20 0.767/0.763 0.777/0.635

Expt. Model Specified Theoretical / Model Simulatedα β d0 h c0

L1U2 1 2 0.2 0.79/0.76 1.06/1.04L1U1 1 1 0.2 0.72/0.68 0.80/0.79L1UM1 1 -1 0.2 0.43/0.39 0.46/0.45L1UM2 1 -2 0.2 0.32/0.29 0.40/0.38L0U2 0 2 0.2 0.72/0.68 0.97/0.95L0UM2 0 -2

Variation in Low-level Shear

α= +1

β=0

α= +1

β=0

Initial Depth = 0.59

Initial Depth = 0.41

Moderate Low-level Shear, Different Initial Cold Pool Depth

Time Averaged Fields

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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

Time Averaged Over 6 Time Units

Fields at T = 12

Time Averaged Streamlines

Solution Highly Non-Steady

c

α= +3

β=0

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Strong Low-level Shear, Different Initial Cold Pool Depth





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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L1U2 1 2 0.2 0.79/0.76 1.06/1.04L1U1 1 1 0.2 0.72/0.68 0.80/0.79L1UM1 1 -1 0.2 0.43/0.39 0.46/0.45L1UM2 1 -2 0.2 0.32/0.29 0.40/0.38L0U2 0 2 0.2 0.72/0.68 0.97/0.95L0UM2 0 -2 0.2 0.25/0.20 0.32/0.30



Variation in Upper-level Shear

Effect of Cold Pool Internal Circulation

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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.

γ=1

γ=-1

Negative Internal Shear

Positive Internal Shear

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γ=1

γ=-1

Negative Internal Shear

Positive Internal ShearT=12

No SignificantCirculation Induced by

Cold Pool

• 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.

Conclusions from Numerical Experiments

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Numerical Simulations of Squall Linesin Support of Our Last Argument

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

Linear Low-level Shear Step Inflow Profile with Zero Vorticty

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-10m/s -15m/s

-18m/s -25 m/s

Tim

e (0

-10

Hou

rs)

Stationary

Constant Speed

Propagation

Step Profile Cases

Line is Quasi-Stationary

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15 m/s Step Inflow Profile

t=5h

t=10hRainfall Rate from 6 to 10 h

x

Tim

e

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

Infl

ow P

rofi

les

θ’

η, Div (shaded)Line-Relative Vectors

RKW Simulation Results

Optimal ConditionNo need for

‘Cold Pool Circulation’or ‘Inflow Shear’

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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, 1995

DFW

X

XB5

00:00 UTC 02:45 UTC

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Observed Pre-Squall Line Wind (U) Profiles, 00 UTC, May 8, 1995

• Average shear normal to the line is nearly zero in the lowest 2 km in both profiles.

Dallas-Fort Worth Okmulgee, OK

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, 1999.

ARPS 48 h Forecast at 6 km ResolutionShown are the Composite Reflectivity and Mean Sea-level Pressure.

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Vertical Cross-Section

• Hardly any cold pool temperature perturbation -> Little buoyancyproduction 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

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Summary Point #1

• RKW theory considers the circulation at the edge of the cold pool (via baroclinicvorticity 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 upper-level front-relative flow plays an equally important role in determining the slope of the updraft.

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Summary Point #3

• 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;

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.

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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.