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Dynamics of Rotation, Propagation, and Splitting
METR 4433: Mesoscale Meteorology Spring 2013 Semester Adapted from Materials by Drs. Kelvin Droegemeier, Frank Gallagher III and Ming Xue; and from Markowski and Richardson (2010) School of Meteorology University of Oklahoma
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Dynamics of Isolated Updrafts
Linear theory is a powerful tool for understanding storm dynamics! It can be used to explain Origin of mid-level rotation Mesocyclone intensification Deviate motion and propagation Nonlinear theory is needed to explain Splitting We’ve looked at these qualitatively and now will apply theory
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Origin of Mid-Level Rotation
We already established that mid-level rotation is a result of the titling by an updraft of horizontal vorticity associated with environmental shear We’re now going to look at theory, which leads us into the concept of streamwise vorticity
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Origin of Mid-Level Rotation
Begin with vertical vorticity equation
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Origin of Mid-Level Rotation
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Origin of Mid-Level Rotation
Now move into a storm-relative reference frame, where C is the storm motion and V-C the storm-relative wind
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Origin of Mid-Level Rotation
Horizontal Vorticity
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Origin of Mid-Level Rotation
Tilting generates vertical vorticity, with the vortex coupled straddling the updraft Once vertical vorticity is present, it can then be advected – with the only wind that matters the STORM-RELATIVE wind!!
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Origin of Mid-Level Rotation
Horizontal Vorticity
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Example of Crosswise Vorticity
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Example of Crosswise Vorticity
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Example of Crosswise Vorticity
Note two updrafts: The “hill,” which is the primary updraft, and the vertical motion induced by storm-relative flow in conjunction with it
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Example of Crosswise Vorticity
The net updraft (black) and vertical vorticity (red). The storm-relative winds are zero at this early stage because the updraft is moving along the hodograph (red dot)
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Example of Streamwise Vorticity
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Example of Streamwise Vorticity
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Example of Streamwise Vorticity
Note two updrafts: The “hill,” which is the primary updraft, and the vertical motion induced by storm-relative flow in conjunction with it
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Example of Crosswise Vorticity
The net updraft (black) and vertical vorticity (red). The storm-relative winds at low-levels are from the south: draw line from red dot (storm motion) back to the hodograph
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Idealized Hodograph Note locations of streamwise and crosswise vorticity depending upon storm-relative winds Storm Motion
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Idealized Hodograph Note locations of streamwise and crosswise vorticity depending upon storm-relative winds Storm-Relative Winds Storm Motion
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Idealized Hodograph Note locations of streamwise and crosswise vorticity depending upon storm-relative winds Storm-Relative Winds Storm Motion
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Streamwise Vorticity It is the vorticity in the direction of the unit vector storm-relative wind The numerator is called the Helicity Density, as noted previously in class
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Relative Helicity The Relative Helicity, or Normalized Helicity Density, is just the streamwise vorticity normalized by the magnitude of the vorticity, or Note that Where theta is the angle between the vorticity and storm-relative velocity vectors
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Relative Helicity Dividing by the magnitude of the vorticity vector yields the relative helicity It’s clear that Relative Helicity is simply the cosine of the angle between the vorticity and storm-relative velocity vectors and varies between -1 and +1
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Optimal Conditions for a Mesocyclone
Optimal conditions for a mesocyclone are Streamwise vorticity (alignment between storm-relative winds and environmental horizontal vorticity) – that is, Relative Helicity close to 1 Strong storm-relative winds BOTH conditions must be met Can quantify these two effects theoretically
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Optimal Conditions for a Mesocyclone
r = correlation coefficient between w and vertical vorticity P is proportional to updraft growth rate
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Optimal Conditions for a Mesocyclone
The cosine term is called the relative helicity (cosine of angle between the storm-relative wind vector and the horizontal vorticity vector). It is the fraction of horizontal vorticity that is streamwise. When cosine term is zero, horizontal inflow vorticity is purely crosswise.
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Optimal Conditions for a Mesocyclone
Note that alignment of the horizontal vorticity vector and storm-relative wind vector is NOT SUFFICIENT. One must have strong storm-relative winds to co-locate updraft and vertical vorticity (via the P term).
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Testing the Theory with a 3D Cloud Model
Droegemeier et al. (1993)
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Testing the Theory with a 3D Cloud Model
Droegemeier et al. (1993)
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Testing the Theory with a 3D Cloud Model
Actual Theoretical Droegemeier et al. (1993)
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Testing the Theory with a 3D Cloud Model
Actual Droegemeier et al. (1993)
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Testing the Theory with a 3D Cloud Model
Actual Droegemeier et al. (1993)
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Testing the Theory with a 3D Cloud Model
Notice how the correlation between vertical velocity and vertical vorticity increases over time as the vorticity and velocity contours begin to overlap. Droegemeier et al. (1993)
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Note the Large Relative Helicity Isn’t Enough – Need Storm Storm-Relative Winds as Well
Droegemeier et al. (1993)
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Testing the Theory with a 3D Cloud Model
The rule of thumb of 90 degrees of turning and at least 10 m/s of storm-relative winds in the 0-3 km layer holds true Droegemeier et al. (1993)
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Updraft Splitting We discussed previously updraft splitting and the role of precipitation, noting that storms split in 3D cloud models even when precipitation is “turned off” Now we look at the dynamics of splitting
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Dynamics of Isolated Updrafts
We want to obtain an expression for p’ = Stuff.... Note that we did this before (spin and splat)!
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Dynamics of Isolated Updrafts
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Dynamics of Isolated Updrafts
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Dynamics of Isolated Updrafts
Can you spot the nonlinear versus linear terms?
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Dynamics of Isolated Updrafts
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Nonlinear Theory of an Isolated Updraft
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Note that low pressure exists at the center of each vortex
and thus “lifting pressure gradients” cause air to rise from high to low pressure, enhancing the updraft beyond buoyancy effects alone and leading to splitting
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Note that low pressure exists at the center of each vortex
and thus “lifting pressure gradients” cause air to rise from high to low pressure, enhancing the updraft beyond buoyancy effects alone and leading to splitting
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Dynamics of Isolated Updrafts
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Nonlinear Theory of an Isolated Updraft
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Nonlinear Theory of an Isolated Updraft
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Selective Enhancement and Deviate Motion of Right-Moving Storm
For a purely straight hodograph (unidirectional shear, e.g., westerly winds increasing in speed with height and no north-south wind present), an incipient supercell will form mirror image left- and right-moving members
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Straight Hodograph: Idealized
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Selective Enhancement and Deviate Motion of Right-Moving Storm
For a curved hodograph, the southern member of the split pair tends to be the strongest It also tends to slow down and travel to the right of the mean wind
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Curved Hodograph – Selective Enhancement of Cyclonic Updraft
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Obstacle Flow – Wrong! Newton and Fankhauser (1964)
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H L Magnus Effect – Wrong! Slow Storm Updraft Fast
Via Bernoulli effect, low pressure located where flow speed is the highest, inducing a pressure gradient force that acts laterally across the updraft Storm Updraft Newton and Fankhauser (1964)
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Dynamics of Isolated Updrafts
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Linear Theory of an Isolated Updraft
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Linear Theory of an Isolated Updraft
This equation determines where pressure will be high and low based upon the interaction of the updraft with the environmental vertical wind shear Rotunno and Klemp (1982)
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Shear = V(upper) – V(lower)
Vertical Wind Shear Up Shear = V(upper) – V(lower) East
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Linear Theory of an Isolated Updraft
x Storm Updraft (w>0) P’>0 P’<0 Rotunno and Klemp (1982)
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Unidirectional Shear (Straight Hodograph)
Note that if the shear vector is constant with height (straight hodograph), the high and low pressure centers are identical at all levels apart from the intensity of w Storm Updraft (w>0) Storm Updraft (w>0) Storm Updraft (w>0) P’>0 P’<0 P’>0 P’<0 P’>0 P’<0 Low Mid Upper Low Mid Upper
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Straight Hodograph S S Rotunno and Klemp (1982)
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Straight Hodograph: Idealized
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Straight Hodograph: Real
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Turning Shear Vector Note that if the shear vector turns with height (curved hodograph), so do the high and low pressure centers P’<0 P’>0 Storm Updraft (w>0) Storm Updraft (w>0) Storm Updraft (w>0) P’>0 P’<0 P’>0 P’<0 Low Mid Upper Mid Upper Low
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Curved Hodograph S S S S Rotunno and Klemp (1982)
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Curved Hodograph – Selective Enhancement of Cyclonic Updraft
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Curved Hodograph – Selective Enhancement of Cyclonic Updraft
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Predicting Thunderstorm Type: The Bulk Richardson Number
Need sufficiently large CAPE (2000 J/kg) Denominator is really the storm-relative inflow kinetic energy (sometimes called the BRN Shear) BRN is thus a measure of the updraft potential versus the inflow potential
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Results from Observations and Models
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General Guidelines for Use
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BRN in the Modeling Study
Droegemeier et al. (1993)
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BRN in the Modeling Study
Droegemeier et al. (1993)
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BRN in the Modeling Study
Droegemeier et al. (1993)
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Supercell Longevity/Predictability
Observations show that supercell storms are relatively long-lived and thus more easily predictable than their weaker-shear, weakly-rotating counterparts 33 min Forecast Low-level Reflectivity Observed Low-level Reflectivity
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Helicity It has been proposed that storms having high helicity (rotating updrafts) are resilient to turbulent decay and thus live longer Consider the 3D vector vorticity equation derived earlier Consider also the idealized situation in which the vector velocity is exactly parallel to the vector vorticity and differs only by a constant (called the abnormality, or lambda)
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Helicity Such a flow, in the absence of baroclinic effects and friction, is called a Beltrami flow – and is purely helical. Under these conditions, it is easy to show that
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Helicity Nonlinear Vorticity Advection Stretching Tilting In a Beltrami flow (and valid for supercells, with caveats), the nonlinear advection exactly cancels stretching plus tilting Because advection and stretching create small scales (cascade), the downscale cascade of energy is effectively blocked, possibly leading to longer-lived storms
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Helicity In fluid dynamics, helicity typically is integrated over a volume. In storm dynamics, the velocity of significance is the storm-relative wind, and thus helicity is not Galilean invariant (depends upon storm motion) We also are concerned about storm inflow, so helicity is typically computed over the inflow layer (0-3 km) and is termed Storm-Relative Environmental Helicity (SREH)
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Storm Relative Environmental Helicity
SREH is the area swept out by the S-R winds between the surface and 3 km It includes all of the key ingredients mentioned earlier It is graphically easy to determine
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Storm Relative Helicity
180 This area represents the 1-3 km helicity 3 km 4 km 2 km 5 km 7 km 6 km 1 km SFC Storm Motion 270 SREH Potential Tornado Strength m2 s-2 Weak m2 s-2 Strong > 450 m2 s-2 Violent
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Helicity Computing helicity from wind data is easy, once storm motion is either known or assumed based upon environmental winds (see also Eq in the text)
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SREH in the Modeling Study
Droegemeier et al. (1993)
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SREH in the Modeling Study
Droegemeier et al. (1993)
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SREH in the Modeling Study
Droegemeier et al. (1993)
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SREH in the Modeling Study
Droegemeier et al. (1993)
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SREH in the Modeling Study
Droegemeier et al. (1993)
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Real Data
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Key Summary Points Streamwise vorticity is a key ingredient in supercell dynamics; however, the alignment between the vertical velocity and vertical vorticity vectors is insufficient – also need strong storm-relative winds and turning of the wind shear vector with height Updraft splitting is principally the result of nonlinear dynamics in the form of lifting pressure gradients on the flanks of the storm Deviate updraft motion is principally the result of linear dynamics in the form of lateral pressure gradient forces associated with the turning of the environmental shear vector with height Helicity is believed responsible for the longevity of supercells The Bulk Richardson Number is a good predictor of storm type Storm-Relative Environmental Helicity is a good predictor of storm type and net updraft rotation, including sign
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