Symmetric, 2-D Squall Line
Tropical Squall Lines:
(Zipser, 1977)
Severe Mid-Latitude
Squall Lines:
(Newton, 1963)
Frontal Squall Lines:
(Carbone, 1982)
Basic Equations: 2D Squall Line
du
¶p
= -C p q v
+ fv + Fx
dt
¶x
⁄
dw
¶p
=- C p q v
+B
dt ¶z
*Also, no vortex
tilting or stretching
Or, more simply, consider the 2D
horizontal vorticity equation:
dh
¶B
=dt
¶x
æ ¶u ¶w ö
where h =
è ¶z ¶x ø
RKW Theory
Rotunno et al.
(JAS, 1988)
“Optimal”
condition for
cold pool lifting
C/∆u > 1
C/∆u = 1
C/∆u < 1
Early System Evolution
“Optimal”
C/∆u << 1
C/∆u ~ 1
Mature System:
C/∆u > 1
2D Convective System Evolution:
C/∆u << 1
C/∆u ~ 1
C/∆u > 1
Weak shear, strong cold pool: rapid evolution
Strong shear, weak cold pool: slow evolution
2D Convective System Evolution:
So, what’s optimal??
C/∆u << 1
C/∆u ~ 1
C/∆u > 1
RKW Theory: all other things being equal
(e.g., same external forcing), squall line
strength/longevity is “optimized” when the
circulation associated with the systemgenerated cold pool remains “in balance”
with the circulation associated with the lowlevel vertical wind shear.
Issue:
Squall-lines are observed to be strong
and long-lived for a wider range of
environments than suggested by the
models (e.g., weaker shears, deeper
shears,….). So, what is the utility of RKW
theory?
Thorpe et al. (1982)
(2D)
Squall Lines steadiest
when shear confined
to low-levels!
Fovell (1988) (2D)
Weisman et al. (1988) (3D)
Weisman et al.
(1988) (3D)
Weisman et al. (1988) (3D)
Weisman and
Rotunno (2004)
Weisman and Rotunno (2004)
Weisman and Rotunno (2004)
Weisman and Rotunno (2004)
Weisman and Rotunno (2004)
Weisman and Rotunno (2004)
Wmax (ms-1) 3-6 h
Total Rainfall 1-6 h
Total Condensation 1-6 h
RKW Theory: all other things being equal
(e.g., same external forcing), squall line
strength/longevity is “optimized” when the
circulation associated with the systemgenerated cold pool remains “in balance”
with the circulation associated with the lowlevel vertical wind shear.
Issue:
Squall-lines are observed to be strong
and long-lived for a wider range of
environments than suggested by the
models (e.g., weaker shears, deeper
shears,….). So, what is the utility of RKW
theory?
Now Consider a 3D Squall
Line….without Coriolis:
du
¶p
= -C p q v
+ fv + Fx
dt
¶x
⁄
dv
¶p
= -C p qv
- fu + Fy
dt
¶y
⁄
dw
¶p
=- C p q v
+B
dt ¶z
20 ms-1 shear, no Coriolis forcing
5 May 1996 18:48 GMT
How can we
systematically produce
the observed line-end
vortex pattern?
Weisman and Davis (1998)
“Optimal”
Mature System:
C/∆u ~ 1
C/∆u > 1
Weisman and Davis (1998)
Weisman and Davis (1998)
Weisman and
Davis (1998)
Weisman and Davis (1998)
Vortex Lines: Us=20 ms-1 over 2.5 km
t=4h
Weisman and Davis (1998)
Line-end vortex mechanisms:
Mature Phase:
Weisman and
Davis (1998)
f=0
Vortex Tube
Circulation:
(
)
C º ò v × dl = ò Ñ ´ v × dA
c
s
dC
dv
=ò
× dl = ò Bk × dl
c
c
dt
dt
Vertical Vorticity:
…flux form
dV
¶w
= w h ×Ñh w + (V + f )
dt
¶z
⁄
¶V
¶ é ¶v
ù
= - ê w + u (V + f ) ú
¶t
¶x ë ¶z
û
¶ é ¶u
ù
- ê -w + v (V + f ) ú
¶y ë
¶z
û
⁄
⁄
Circulation:
¶C
¶V
æ ¶V ö ˆ
= òò ds = ò ç ÷ × ndl
è
ø
¶t
¶t
¶t
A
Weisman and
Davis (1998)
f=0
b
¶C
é ¶u
ù
= ò ê -w + vV údx + ò fv × nˆ dl
¶t a ë
¶z
û
⁄
⁄
…tilting of system-generated
horizontal vorticity
Rear-inflow jet
(Davis and Weisman, 1994; Weisman and
Davis, 1998; Davis and Galarneau, 2009)
Role of Line-End Vortices
Focuses and
Intensifies
Rear-Inflow
Jet
Now Consider a 3D Squall
Line….with Coriolis:
du
¶p
= -C p q v
+ fv + Fx
dt
¶x
dv
¶p
= -C p qv
- fu + Fy
dt
¶y
dw
¶p
=- C p q v
+B
dt ¶z
20 ms-1 shear, Coriolis forcing
dV
¶w
= w h ×Ñh w + (V + f )
dt
¶z
¶V
é ¶w ¶v ¶w ¶u ù
é ¶u ¶v ù
+ v ×ÑV = ê +
- (V + f ) ê + ú
ú
¶t
ë ¶x ¶z ¶y ¶z û
ë ¶x ¶y û
¶V
¶ é ¶v
ù ¶ é ¶u
ù
= - ê w + u ( f + V ) ú - ê -w + v ( f + V ) ú
¶t
¶x ë ¶z
¶z
û ¶y ë
û
¶C
¶V
æ ¶V ö ˆ
= òò ds = ò ç ÷ × ndl
è ¶t ø
¶t
¶t
A
¶V
¶ é ¶v
ù
= - ê w + u (V + f ) ú
¶t
¶x ë ¶z
û
¶ é ¶u
ù
- ê -w + v (V + f ) ú
¶y ë
¶z
û
b
¶C
é ¶u
ù
= ò ê -w + vV údx + ò fv × nˆ dl
¶t a ë
¶z
û