Chapter 2: basic equations and tools
All sections are considered core material, but the emphasis is on the
“new” elements in the last three sections:
•
•
•
2.5 – pressure perturbations
2.6 – thermodynamic diagrams
2.7 - hodographs
flow vs vorticity: the right-hand rule
𝜂 =𝑗∙𝛻×𝑣
hydrostatic pressure perturbations
2004 – 2046 UTC
2049 – 2130 UTC
2134-2214 UTC
H
H
L
H
L
H
L
L
L
H
H
H
Measured height (1-m increments) of the 595-hPa surface during a winter storm
over the Med Bow Range Wyoming (11 February 2008)
(Parish and Geerts 2013)
hydrostatic pressure perturbations:
stratified flow over an isolated peak
wind
H
L
H
L
hydrostatic pressure perturbations:
2D stratified flow over a mountain
Fig. 6.6: polarization relations between q’, p’, u’, and w’ in a
westward tilting, vertically-propagating internal gravity
wave
blue = cold, red = warm
wind U
cg
H
c
terrain
L
H
examine column T’ above
The pressure perturbation field in a buoyant bubble (e.g. a Cu tower)
contains both a hydrostatic and a non-hydrostatic component.
B>0
B<0
the buoyancy-induced pressure perturbation gradient acceleration
(BPPGA):
  pB'
z
narrow buoyant blob
x
Analyze:
B
 p  F B
z
2
'
B
wide buoyant layer
Shaded area is
buoyant B>0
This is like the Poisson eqn in electrostatics, with FB the charge density, p’B the electric
potential, and p’B show the electric field lines.
B 2
L
The + and – signs indicate highs and lows: p  
z
'
B
where L is the width of the buoyant parcel
BPPGA   pB'
Where p’B>0 (high), 2p’B <0, thus the divergence of [- p’B] is
positive, i.e. the BPPGA diverges the flow, like the electric field.
 The lines are streamlines of BPPGA, the arrows indicate the
direction of acceleration.
assume _ hydrostatic _ balance
Dw
1 pB'
i.e.

B0
Dt
 z
then
'
2 '
1

p


B

pB
B
 Within the buoyant parcel, the BPPGA always opposes the buoyancy,
 B or

 z
z
z 2
thus the parcel’s upward acceleration is reduced.
Now because
 A given amount of B produces a larger net upward acceleration in a
 B
smaller parcel
 2 pB' 
z
Proof
 for a very wide parcel, BPPGA=B
we conclude that
(i.e. the parcel, though buoyant, is hydrostatically balanced)
(in this case the buoyancy source equals d2p’/dz2)
 2 pB'  2 pB'

0
x 2
y 2
This pressure field contains both a hydrostatic and a
non-hydrostatic component.
1 p'h
B
 z
B>0
 2 pB' 
B
z
p’B
explanation: the image on the right shows buoyantly-induced (p’b) perturbation
pressure field with a high above and a low below the warm core. The spreading of the
isobars in the warm core suggest that these pressure perturbations are at least partly
hydrostatic: greater thickness in the warm core. Yet a pure hydrostatic component
(p’h) would just have a low below the warm core, down to the surface (Fig. 2.7).
interpreting pressure perturbations
Note that p’ = p’h+p’nh = p’B+p’D
1 p'h
and p’nh = p’-p’h
B
 z
B
p’B is obtained by solving  2 p B'  F B
with
z
p’D = p’-p’B
p’h is obtained from
B
 0 at top and bottom.
z
Pressure perturbations in a density current
pressure units: (Pa)
L
H
L
H
H
L
L
H
H
Fig. 2.6
interpretation: use
Bernoulli eqn along a
streamline
2
v
2

p'

 Bz  const.
Pressure perturbations in a buoyant bubble, e.g. a growing cumulus
H
pressure units: (Pa)
L
L
L
L
L
-250
+225
H
H
H
L
L
2K bubble, radius = 5 km, depth 1.5
km, released near ground in an
environment with CAPE=2200 J/kg.
Fields are shown at t=10 min
L
H
Fig. 2.7
Cumulus bubble observation
Example of a growing cu on Aug. 26th, 2003 over Laramie. Two-dimensional velocity
field overlaid on filled contours of reflectivity (Z [dBZ]); solid lines are selected
streamlines. (source: Rick Damiani)
Cumulus bubble observation
dBZ
20030826, 18:23UTC
8m/s
•
Two counter-rotating vortices are visible in the ascending
cloud-top.
•
They are a cross-section thru a vortex ring, aka a toroidal
circulation (‘smoke ring’)
(Damiani et al., 2006, JAS)
Shear interacting with an updraft
 vh  u v 
S
 , 
z  z z 
ambient wind
p’D> 0 (a high or “H”) on the upshear side of a convective updraft, and
p’D< 0 (L) on the downshear side
Shear, buoyant updraft, and linear dynamic pressure gradient
S
H
Shear deforms the parcel in the opposite direction as
that due to a convective updraft on the upshear side of
that updraft. In other words, the respective horizontal
vorticity vectors point in opposite directions on the
upshear side, yielding an erect upshear flank. The
downshear flank is tilted downwind because the
respective vorticity vectors supplement each other.
Fig. 2.8
L
skew T log p
CAPE,
and CIN
always use virtual temperature !
radiosonde analysis
model sounding analysis
Fig. 2.9
limitations of parcel theory (section 3.1.2)
• 𝑤𝑚𝑎𝑥 = 2𝐶𝐴𝑃𝐸
• factors limiting updraft strength and controlling Cu width:
– downward BPPGF … increases with Cu width
– entrainment of dry ambient air … undiluted core vanishes faster in
more narrow Cu
– hydrometeor loading … reduces B directly
downdraft CAPE
Fig. 2.10
hodographs
c : storm motion
vr= v-c : storm-relative mean flow
S: mean shear
wh : mean horizontal vorticity
wh = ws+ wc : streamwise & crosswise vorticity
Fig. 2.11 and 2.12
horizontal vorticity
shear vector S
.M
c

1
M 
 (u , v )dz
 o h 0
h
mass-weighted deep-layer mean wind
storm-relative flow vr
horizontal vorticity
 w v u w  ˆ 
wh   ,     ,    k  S
 y z z x 
 w   w 
 u   v 
assume O   , O    O  , O  
 x   y 
 z   z 

directional shear
vs. speed shear
x storm
motion c
Fig. 2.13
real hodographs near observed severe storms
Fig. 2.14
•
•
 


vr  w h
ws  wh cos 
streamwise ws  
vr
  


vr v r  w h 
w


w

w
cross-wise


c
c
h sin 
vr
vr

streamwise vorticity
definition of helicity (Lilly 1979)


top
top
top
 
 
 
 ˆ 
ˆ
H   vr ws dz   vr  wh dz   vr  (k  S )dz   k  S  vr dz
top
0
0
0
0
• the top is usually 2 or 3 km (low level !)
• H is maximized by high wind shear NORMAL to the stormrelative flow
–  strong directional shear
• H is large in winter storms too, but static instability is missing
storm-relative flow
horizontal vorticity
streamwise vorticity produces helical flow
Fig. 2.15
more on this in chapter 7