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

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

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

Origin of Mid-Level Rotation Begin with vertical vorticity equation

Origin of Mid-Level Rotation

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

Origin of Mid-Level Rotation Horizontal Vorticity

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!!

Origin of Mid-Level Rotation Horizontal Vorticity

Example of Crosswise Vorticity

Example of Crosswise Vorticity

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

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)

Example of Streamwise Vorticity

Example of Streamwise Vorticity

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

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

Idealized Hodograph Note locations of streamwise and crosswise vorticity depending upon storm-relative winds Storm Motion

Idealized Hodograph Note locations of streamwise and crosswise vorticity depending upon storm-relative winds Storm-Relative Winds Storm Motion

Idealized Hodograph Note locations of streamwise and crosswise vorticity depending upon storm-relative winds Storm-Relative Winds Storm Motion

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

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

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

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

Optimal Conditions for a Mesocyclone r = correlation coefficient between w and vertical vorticity P is proportional to updraft growth rate

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.

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).

Testing the Theory with a 3D Cloud Model Droegemeier et al. (1993)

Testing the Theory with a 3D Cloud Model Droegemeier et al. (1993)

Testing the Theory with a 3D Cloud Model Actual Theoretical Droegemeier et al. (1993)

Testing the Theory with a 3D Cloud Model Actual Droegemeier et al. (1993)

Testing the Theory with a 3D Cloud Model Actual Droegemeier et al. (1993)

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)

Note the Large Relative Helicity Isn’t Enough – Need Storm Storm-Relative Winds as Well Droegemeier et al. (1993)

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)

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

Dynamics of Isolated Updrafts We want to obtain an expression for p’ = Stuff.... Note that we did this before (spin and splat)!

Dynamics of Isolated Updrafts

Dynamics of Isolated Updrafts

Dynamics of Isolated Updrafts Can you spot the nonlinear versus linear terms?

Dynamics of Isolated Updrafts

Nonlinear Theory of an Isolated Updraft

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

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

Dynamics of Isolated Updrafts

Nonlinear Theory of an Isolated Updraft

Nonlinear Theory of an Isolated Updraft

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

Straight Hodograph: Idealized

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

Curved Hodograph – Selective Enhancement of Cyclonic Updraft

Obstacle Flow – Wrong! Newton and Fankhauser (1964)

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)

Dynamics of Isolated Updrafts

Linear Theory of an Isolated Updraft

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)

Shear = V(upper) – V(lower) Vertical Wind Shear Up Shear = V(upper) – V(lower) East

Linear Theory of an Isolated Updraft x Storm Updraft (w>0) P’>0 P’<0 Rotunno and Klemp (1982)

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

Straight Hodograph S S Rotunno and Klemp (1982)

Straight Hodograph: Idealized

Straight Hodograph: Real

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

Curved Hodograph S S S S Rotunno and Klemp (1982)

Curved Hodograph – Selective Enhancement of Cyclonic Updraft

Curved Hodograph – Selective Enhancement of Cyclonic Updraft

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

Results from Observations and Models

General Guidelines for Use

BRN in the Modeling Study Droegemeier et al. (1993)

BRN in the Modeling Study Droegemeier et al. (1993)

BRN in the Modeling Study Droegemeier et al. (1993)

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

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)

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

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

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)

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

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 150 - 300 m2 s-2 Weak 300 - 500 m2 s-2 Strong > 450 m2 s-2 Violent

Helicity Computing helicity from wind data is easy, once storm motion is either known or assumed based upon environmental winds (see also Eq. 8.15 in the text)

SREH in the Modeling Study Droegemeier et al. (1993)

SREH in the Modeling Study Droegemeier et al. (1993)

SREH in the Modeling Study Droegemeier et al. (1993)

SREH in the Modeling Study Droegemeier et al. (1993)

SREH in the Modeling Study Droegemeier et al. (1993)

Real Data

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