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the dynamics of convection
1. Cumulus cloud dynamics
•
The basic forces affecting a cumulus cloud
–
–
–
–
buoyancy (B)
buoyancy-induced pressure perturbation gradient acceleration (BPPGA)
dynamic sources of pressure perturbations
entrainment: Simple models of entraining cumulus convection (E)
Suggested readings:
M&R Section 2.5: fundamentals of convection and buoyancy
M&R Section 2.6: pressure perturbations
Bluestein (Synoptic-Dynamic Met Pt II, 1993) Part III
Houze (Cloud Dynamics, 1993), Chapter 7
Holton (Dynamic Met, 2004) sections 9.5 and 9.6
Cotton and Anthes (Storm and Cloud Dynamics, 1982)
Emanuel (Atmospheric Convection, 1994)
Essential Role of Convection
• You have learned about the role of baroclinic systems in the
atmosphere: they transport sensible heat and water vapor
poleward, offsetting a meridional imbalance in net radiation.
• Similarly, thermal convection plays an essential role in the vertical
transport of heat in the troposphere:
– the vertical temperature gradient that results from radiative
equilibrium exceeds that for static instability, at least in some regions
on earth.
Cumulus Clouds
•
Range in size from
–
–
–
–
–
cumuli (less than 1000 m in 3D diameter)
congesti (~1 km wide, topping near 4-5 km)
cumulonimbus clouds (~10 km) to
thunderstorm clusters (~100 km) to
mesoscale convective complexes (~500 km).
topping near the tropopause
•
•
All are ab initio driven by buoyancy B
vertical equation of motion:
Dw
1 p'

B
Dt
 z buoyancy
PPGF
•
force
PPGF: pressure perturbation gradient force
buoyancy:
we derive buoyancy from vertical momentum conservation equation
Dw  1 p

g
vertical equation of motion, non-hydrostatic
Dt
 z
p ( x, y , z , t )  p ' ( x, y , z , t )  p ( z )
  1   ' ( x, y , z , t )   ( z )

assume : (1) p '  p  '   ; (2)
dp
  g
dz
basic state is
hydrostatically balanced
then
Dw
p
p
p '
p '
p '
p
p '
'
 
 g  ' 
'
 
  '  
g
Dt
z
z
z
z
z
z
z

terms cancel
small term

c p'
B
 ' Tv' p'
     qh   0.61qv'  v
 qh
g
cp p
 Tv p

'
define buoyancy B:
basic relations :
p  Rd Tv
p 
  T  o 
 p
Rd
cp
Tv  T (1  0.61qv )
c p  cv  Rd
scaling the buoyancy force
•
Complete expression for buoyancy:
 '

c
p
'
'
v
B  g   0.61qv 
 qh 


cp p


see (Houze p. 36)
where qh is the mixing ratio of condensed-phase water (‘hydrometeor loading’).
c p'
B
  '  0.61 qv'  v   qh
g
cp p
In terms of its effect on buoyancy (B/g), 1 K of excess heat ‘ is
equivalent to ...
... 5-6 g/kg of water vapor (positive buoyancy)
... 3-4 mb of pressure deficit (positive buoyancy)
... 3.3 g/kg of water loading (negative buoyancy)
This shows that ’ dominates. All other effects can be significant under
some conditions in cumulus clouds.
buoyancy:
outside of cloud, buoyancy is proportional to the virtual pot temp perturbation
Dw  1 p
g

 z
Dt
1 p '
Dw
B

 z
Dt
'

c p'
'
B
 qh
    0.61qv'  v
cp p
g
 
outside of cloud (qh  0) :
B  v' 
   0.61qv'
g v 
'
The buoyancy force
•
To a first order, the maximum
updraft speed can be estimated
from sounding-inferred CAPE:
this ignores effect of water vapor (+B)
and the weight of hydrometeors (-B).
Dw
w 1 w2
'
w

Bg
Dt
z 2 z

thus
LNB
w  2  Bdz 
LFC
2g

LNB
  ' dz 
2CAPE
LFC
w= sqrt(2CAPE) is the thermodynamic updraft strength limit
This updraft speed is a vast over-estimate, mainly b/o two opposing forces.
•
pressure perturbations: Buoyancy-induced (or ‘convective’) ascent of an air parcel
•
entrainment
disrupts the ambient air. On top of a rising parcel, you ‘d expect a high (i.e. a positive
pressure perturbation), simply because that rising parcel pushes into its
surroundings. The resulting ‘perturbation’ pressure gradient enables compensating
lateral and downward displacement as the parcel rises thru the fluid. Solutions show
the compensating motions decaying away from the cloud, concentrated within about
one cloud diameter.
this pressure field contains both a hydrostatic
and a non-hydrostatic component
B>0
B<0
Fig. 2.2 in M&R
discuss pressure changes in a hydrostatic
atmosphere (M&R 2.6.1)
• mass conservation:
– pressure tendency = vertically integrated mass divergence
• hypsometric eqn:
– pressure tendency = vertically integrated temperature change

 
 

v  1
2nd force: pressure perturbation

p ' Bk  v  v  fk  v
t

gradient acceleration (PPGA)
multiply _ by _  , take _  



   v
B
(M&R section 2.6.3)
  2 p '
    v  v    f
t
z
now

 2 psyn '   f is the synoptic scale pressure field
   v  0 anelastic continuity eqn
note  2 psyn '   psyn '


B
2
 p' 
    v  v    f
psyn '  0 in cyclonic circulatio n (  0)
z
psyn '  0 in anticyclonic circulatio n (  0)
thus, ignoring the Coriolis force,
 2 p '  FB  FD FB: buoyancy source
partition : p '  pB'  pD'
where
 2 pB' 
B
z
tensor
notation
v j 


 
 p     v  v   

v
 i

x j 
xi 
2
'
D
Dw
1 pB' 1 pD'


B
Dt
 z  z
FD: dynamic source
** These equations are fundamental to
understand the dynamics of convection,
ranging from shallow cumuli to isolated
thunderstorms to supercells.
For now we focus on FB.
Later, we ‘ll show that FD is essential to
understand storm splitting and storm
motion aberrations.
It can be shown that


 2 pD'     v  v 
  2  2  2

2  D  
2
  u v w 
with    , ,  (divergence vector)
 x y z 


  w v u w v u 
with D  
 , 
,   (deformation vector)
 y z z x x y 
  w v u w v u 
with   
 , 
,   (vorticity vector)
 y z z x x y 
2  2  2
'
 pD   2  D  


Also, p’D> 0 (a high H) on the upshear
side of a convective updraft, and p’D< 0
(a low L) on the downshear side
the buoyancy-induced pressure perturbation gradient acceleration
(BPPGA):
  pB'
z
x
Analyze:
B
 p  F B
z
2
'
B
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 pB>0 (high), 2pB <0, thus the divergence of [- pB] is positive,
i.e. the BPPGA diverges the flow, like the electric field.
assume _ hydrostatic _ balance
Dw
1 pB'
i.e.

B0
Dt
 z
 The lines are streamlines of BPPGA, the arrows indicate the
then
direction of acceleration.
1 pB'
B
 Within the buoyant parcel, the BPPGA always opposes the buoyancy,  z
or
thus the parcel’s upward acceleration is reduced.
 B  2 pB'


z
z 2
 A given amount of B produces a larger net upward acceleration in a
because
smaller parcel
 B
 2 pB' 
z
 for a very wide parcel, BPPGA=B
this _ implies _ that
(i.e. the parcel, though buoyant, is hydrostatically balanced)
 2 pB'  2 pB'
(in this case the buoyancy source equals d2p’/dz2)

0
2
2
x
y
(10 km)
t=13 min
(3 km)
t=8 min
H
H
L
L
L
Fig. 2.3 in M&R
(no entrainment)
pressure field in a
density current
(M&R, Fig. 2.5)
L
H
H
pressure units: (Pa)
L
L
H
L
H
H
H
2
v
2
Note that p’ = p’h+p’nh = p’B+p’D
p’h is obtained from
interpretation: use
Bernoulli eqn along a
streamline
1 p'h
 B and p’nh = p’-p’h
 z

p'

 Bz  const.
B with B  0 at top and bottom.
p’B is obtained by solving  2 p '  F 
B
B
z
p’D = p’-p’B
z
pressure field in a cumulus cloud
(M&R, Fig. 2.6)
H
pressure units: (Pa)
2K bubble, radius = 5 km, depth 1.5
km, released near ground in
environment with CAPE=2200 J/kg.
Fields shown at t=10 min
L
Note that p’ = p’h+p’nh = p’B+p’D
L
p’h is obtained from
and p’nh = p’-p’h
H
1 p'h
B
 z
p’B is obtained by solving  2 p B' 
H
L
L
with
H
L
H
B
z
B
 0 at top and bottom.
z
p’D = p’-p’B
Third force (also holding back buoyancy):
•
entrainment
entrainment does two things:
(a) both the upward momentum and the buoyancy of a parcel are dissipated by mixing
(b) cloudy air is mixed with ambient dry air, causing evaporation
 L Dq H 
D
L Dq

 w( parcel   env )  

Dt
c p Dt
 c p Dt  mixing
•
•
No elegant mathematical formulation exists for entrainment E. The reason is that
we are entering the realm of turbulence. We are reduced to some simple
conceptual models of cumulus convection.
–
–
Thermals or Bubbles
Plumes or Jets
This is a general expression for continuous, homogenous entrainment (Houze p.
227-230) (1D, steady state):
(assumed)
simplify to 1D,
steady state & solve
Dw
1 dpB' 1 dpD'
 B

E
Dt
 dz  dz
thermal : E  kw
plume : E  w2
1 dme

m dz
 w2 
d 
'
2 
dw
1 dp

w

 B
 w 2
dz
dz
 dz
effect of entrainment on a skew T-log p diagram
M&R Figure 2.4. A possible trajectory (dashed) that might be followed by an updraft
parcel on a skew T-log p diagram as a result of the entrainment of environmental air.
Thermals or Bubbles
• Laboratory studies
• negatively buoyant, dyed parcels are released, with small
density difference relative to the environmental fluid
• basic circulations look like this:
The thermal grows as air is entrained
into the thermal, via:
• turbulent mixing at the leading edge;
• laminar flow into the tail of the thermal
At first vorticity is distributed
throughout the thermal.
Later it becomes concentrated
in a vortex ring.
Results: shape oblate, nearly spherical; volume=3R3
•
•
•
note shear instability along
leading boundary
entrainment rate seems
small at first
undilute core persists for
some time, developing into
a vortex ring
Sanchez et al 1989
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)
2.7 sounding analysis
• CAPE
• CIN
• DCAPE (D for downdraft)
2.8 hodographs
•
•
•
•
total wind vh
 vh  u v 
shear vector S S 
 , 
z  z z 
storm motion c
storm-relative wind vr = vh-c
S
vh
storm-relative flow vr
c
height AGL (km)
2.8.5 true & storm-relative wind near a supercell storm
c
real example
hypothetical profiles:
different wind vh, but identical vr
2.8.6 horizontal vorticity
storm-relative flow
horizontal vorticity
 v w w u  ˆ 
h   ,     ,    k  S
 z y x z 
 w   w 
 u   v 
assume O   , O    O  , O  
 x   x 
 z   z 

•
•
 


vr   h
s  h cos 
streamwise s  
vr
  


vr vr   h 






cross-wise


c
c
h sin 
vr
vr

streamwise vorticity
note error in book
error
definition of helicity (Lilly 1979)


top
top
top
 
 
 
 ˆ 
ˆ
H   vr s dz   vr  h 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