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
1
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
2
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
3
Origin of Mid-Level Rotation
•
Begin with vertical vorticity equation
4
Origin of Mid-Level Rotation
5
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
6
Horizontal
Vorticity
Origin of Mid-Level Rotation
7
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!!
8
Horizontal
Vorticity
Origin of Mid-Level Rotation
9
Example of Crosswise Vorticity
10
Example of Crosswise Vorticity
11
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
12
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)
13
Example of Streamwise Vorticity
14
Example of Streamwise Vorticity
15
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
16
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
17
Idealized Hodograph
•
Note locations of streamwise and crosswise vorticity
depending upon storm-relative winds
Storm Motion
18
Idealized Hodograph
•
Note locations of streamwise and crosswise vorticity
depending upon storm-relative winds
Storm-Relative
Winds
Storm Motion
19
Idealized Hodograph
•
Note locations of streamwise and crosswise vorticity
depending upon storm-relative winds
Storm-Relative
Winds
Storm Motion
20
(V  C )    V (V  C )  
s  (V  C )    V
 (V  C )  
s  | V  C |
| V  C |
|V  C |
|V  C |
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
21
(V  C )    V (V  C )  
s  (V  C )    V
 (V  C )  
s  | V  C |
| V  C |
|V  C |
|V  C |
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 stormrelative velocity vectors
22
(V  C )    V (V  C )  
s  (V  C )    V
 (V  C )  
s  | V  C |
| V  C |
|V  C |
|V  C |
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
23
Optimal Conditions for a
Mesocyclone
•
Optimal conditions for a mesocyclone are
• Streamwise vorticity (alignment between stormrelative 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
24
Optimal Conditions for a
Mesocyclone
•
•
r = correlation coefficient between w and
vertical vorticity
P is proportional to updraft growth rate
25
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.
26
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).
27
Testing the Theory with a 3D Cloud Model
Droegemeier et al. (1993)
28
Testing the Theory with a 3D Cloud Model
Droegemeier et al. (1993)
29
Actual
Testing the Theory with a 3D Cloud Model
Theoretical
Droegemeier et al. (1993)
30
Actual
Testing the Theory with a 3D Cloud Model
Droegemeier et al. (1993)
31
Actual
Testing the Theory with a 3D Cloud Model
Droegemeier et al. (1993)
32
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)
33
Note the Large Relative Helicity Isn’t
Enough – Need Storm Storm-Relative
Winds as Well
Relative Helicity
Droegemeier et al. (1993)
34
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)
35
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
36
Dynamics of Isolated Updrafts
We want to obtain an expression for p’ = Stuff....
37
Dynamics of Isolated Updrafts
38
Dynamics of Isolated Updrafts
39
Dynamics of Isolated Updrafts
Can you spot the nonlinear versus linear terms?
40
Dynamics of Isolated Updrafts
41
Nonlinear Theory of an Isolated
Updraft
42
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
43
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
44
Dynamics of Isolated Updrafts
45
Nonlinear Theory of an Isolated
Updraft
46
Nonlinear Theory of an Isolated
Updraft
47
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
48
Straight Hodograph: Idealized
49
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
50
Curved Hodograph – Selective
Enhancement of Cyclonic Updraft
51
Obstacle Flow – Wrong!
Newton and Fankhauser (1964)
52
Magnus Effect – Wrong!
Slow
H
Storm
Updraft
L
Fast
Via Bernoulli effect,
low pressure located where
flow speed is the highest, inducing a pressure gradient force
that acts laterally across the updraft
Newton and Fankhauser (1964)
53
Dynamics of Isolated Updrafts
54
Linear Theory of an Isolated Updraft
55
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)
56
Vertical Wind Shear
Up
Shear = V(upper) – V(lower)
Venv
S
z
East
57
Linear Theory of an Isolated Updraft
y
p 
Venv
w
z
x
P’>0
w
Storm
Updraft
(w>0)
P’<0
Venv
S
z
w
Rotunno and Klemp (1982)
58
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
P’>0 Updraft
(w>0)
P’<0
Low
Low
Storm
P’>0 Updraft P’<0
(w>0)
Storm
P’>0 Updraft P’<0
(w>0)
Mid
Upper
Mid
Upper
59
Straight Hodograph
S
S
Rotunno and Klemp (1982)
60
Straight Hodograph: Idealized
61
Straight Hodograph: Real
62
Turning Shear Vector

Note that if the shear vector turns with height (curved
hodograph), so do the high and low pressure centers
P’<0
Storm
Updraft
(w>0)
P’>0
Storm
P’>0 Updraft P’<0
(w>0)
Storm
Updraft
(w>0)
Mid
P’<0
Upper
P’>0
Low
Mid
Low
Upper
63
S
S
Curved Hodograph
S
S
Rotunno and Klemp (1982)
64
Curved Hodograph – Selective
Enhancement of Cyclonic Updraft
65
Curved Hodograph – Selective
Enhancement of Cyclonic Updraft
66
Predicting Thunderstorm Type: The Bulk
Richardson Number
CAPE
BRN 
2
S
1
where S  (u 6000  u 500 ) 2
2
2
n
n
n
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
67
Results from Observations
and Models
68
General Guidelines for Use
Supercells for 5  BRN  50
Multicells for 35 BRN  400
69
BRN in the Modeling Study
Droegemeier et al. (1993)
70
BRN in the Modeling Study
Droegemeier et al. (1993)
71
BRN in the Modeling Study
Droegemeier et al. (1993)
72
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
73
(V  C )    V (V  C )  
s  (V  C )    V
 (V  C )  
s  | V  C |
| V  C |
|V  C |
|V  C |
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)
74
(V  C )    V (V  C )  
s  (V  C )    V
 (V  C )  
s  | V  C |
| V  C |
|V  C |
|V  C |
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
75
(V  C )    V (V  C )  
s  (V  C )    V
 (V  C )  
s  | V  C |
| V  C |
|V  C |
|V  C |
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
76
(V  C )    V (V  C )  
s  (V  C )    V
 (V  C )  
s  | V  C |
| V  C |
|V  C |
|V  C |
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)
77
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
78
Storm Relative Helicity
180
This area represents
the 1-3 km helicity
3 km
2 km
4 km
5 km
1 km
SFC
6 km
7 km
Storm Motion
270
SREH
Potential Tornado Strength
150 - 300 m2 s-2
Weak
300 - 500 m2 s-2
Strong
> 450 m2 s-2
Violent
79
(V  C )    V (V  C )  
s  (V  C )    V
 (V  C )  
s  | V  C |
| V  C |
|V  C |
|V  C |
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)
80
SREH in the Modeling Study
Droegemeier et al. (1993)
81
SREH in the Modeling Study
Droegemeier et al. (1993)
82
SREH in the Modeling Study
Droegemeier et al. (1993)
83
SREH in the Modeling Study
Droegemeier et al. (1993)
84
SREH in the Modeling Study
Droegemeier et al. (1993)
85
Real Data
86
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
87