PRECIPITATION PROCESSES AT FRONTS
POSSIBLE CONDITIONS PRESENT AT FRONT
1. Air ahead of the front is stable to all forms of instability
Forcing mechanism for vertical motion: ageostrophic circulation
associated with frontogenesis
2. Air is potentially or conditionally unstable
Lifting of air to saturation by ageostrophic circulation triggers convection
3. Air is stable to potentially or conditionally unstable to upright
displacement but unstable to slantwise displacement
Lifting of air to saturation by ageostrophic circulation triggers slantwise
convection
Frontogenesis
As we have learned from the
previous analysis of the SE
equation:
Frontogenesis leads to a direct
circulation
Warm (moist) air rises on the
warm side of the front, leading
to widespread clouds and
precipitation
HOW DO INSTABILITES
MODULATE THIS
PRECIPITATION?
Instabilites:
Convective instability: Body force is gravity, buoyancy acts opposite gravity
and parcels accelerate vertically
(1)
(2)
(3)
(4)
dw
1 p

g
dt
 z
vertical momentum equation
Assume base state density  z  is a function of
height and and perturbation is given by  
   z    x, y, z, t 
Assume base state pressure pz  is a function of
height and and perturbation is given by p
p  pz   px, y, z, t 
p
  g
z
Base state is in hydrostatic balance
1
  

Put (2), (3), (4) into (1), approximate 1    1 
, approximate




do some algebra and get:

dw
1 p  

 g
dt
 z 
    dp dp
1  

  dz dz

Vertical accelerations result from imbalances between the
vertical perturbation pressure gradient force and buoyancy
dw
1 p  

 g
dt
 z 
Assume for for rising parcel, environmental
pressure instantly adjusts to parcel movement
(atmosphere is everywhere hydrostatic (p = 0)
v v
dw

 
  g 
g 
g
dt


v
real
atmosphere
H
L
v = virtual potential temperature
A parcel’s stability can be determined by displacing it vertically a small distance z,
d v
z ,
dz
and realizing that the parcel virtual potential temperature will be conserved v z   v0
assuming that the environmental virtual potential temperature at z is  v   v 0 
d v 
dw
g 
g d v
 v 0   v 0 

z   
z
dt  v 
dz 
 v dz
From this equation, we obtain the criteria for gravitational stability in an unsaturated environment:
d v
 0 stable
dz
d v
0
dz
neutral
d v
 0 unstable
dz
Condensation makes the stability problem considerably more
complicated. In the interest of time, I will state the stability criteria
for moist adiabatic vertical ascent (see Holton p.333, Bluestein’s
books or other books for details):
d v
0
dz
Absolute
instability
Conditional
Instability (CI)
(parcel)
Potential
Instability (PI)
(layer)
 Lv qvs 

  exp

 c pT 
 Lv qvs 

 e  exp

 c pTLCL 
*
e
Definitions:
d e*
0
dz
d e
0
dz
Saturation equivalent potential temperature
Equivalent potential temperature
Example of Potential Instability
Stable sounding to parcel ascent
Lift layer between 1 and 1.25 km
one km in altitude
AIR DESTABILIZES
Synoptic environment conducive to the
development of potential instability
Inertial instability: Body force is centrifugal acceleration due to Coriolis effect,
parcels accelerate horizontally
dv


 fu
dt
y
du


 fv
dt
x
Assume a base state flow that is geostrophic
(1)
dv
 f u g  u 
dt
(2)
horizontal momentum equations
1 
ug  
f y

0
x
du
 fv
dt
Assume a parcel moving at geostrophic base state velocity is displaced across stream
(3)
(4)
u y0  y   ug  y0   f y
u g  y0  y   u g  y0  
u g
y
Parcel conserves its absolute angular momentum
y
Geostrophic wind at location y + y
Put (3) and (4) into (1)
u g 

dv
y
  f  f 
dt
y 

Equation governing inertial instability
u g 

dv
y
  f  f 
dt
y 

f
To understand inertial instability
Consider this simple example
u g
y
 f   g   ga
= absolute geostrophic vorticity
If the absolute vorticity is negative, a parcel of
air when displaced in a geostrophically balanced
flow will accelerate away from its initial position

Inertially stable
300 mb height
field in the vicinity
of a jetstream
COR > PGF
COR= PGF
COR= PGF
COR> PGF
Inertially unstable
 +8  
For a parcel displaced north of jet axis, f
For a parcel displaced south of jet axis, f
u g
is positive while u g is negative.
is positive while
dv
Therefore
is negative and parcel will
dt
If f exceeds the geostrophic shear y ,
y
return to its original position.
y
is positive.
u g
dv
is negative and parcel will accelerate
dt
away from its original position.
Instability summary
In an atmosphere characterized by a hydrostatic and geostrophic base state:
Vertical displacement
Conditional
Instability
d e*
0
dz
Horizontal displacement
Inertial
Instability
g  f 0
Or if we define the absolute geostrophic momentum as m  ug  fy
m u g
so that

 f   g  f 
y
y
Vertical displacement
Conditional
Instability
Momentum
equations
d e*
0
dz
dw g
  v   v 
dt  v
Horizontal displacement
Inertial
Instability
m
0
y
dv
  f m  mg 
dt
What happens if a parcel of air is displaced slantwise in
an atmosphere that is inertially and convectively stable?
Starting point
Let’s assume:
1) We have an east-west oriented front with cold air to the north.
2) The base state flow in the vicinity of the front is in hydrostatic and
geostrophic balance
3) No variations occur along the front in the x (east-west) direction
4) We consider the stability of a tube of air located parallel to the x
axis (east-west oriented tube)
m surfaces and  surfaces in a barotropic and baroclinic environment
m1
m2
m3
m4
m5
m only a function of f along y direction
5
4
3
2
1
p
y
x
Barotropic Atmosphere (no temperature gradient)
5
4
3
2
1
m1
m2
m3
m4
m5
Because of temperature gradient
geostrophic wind increases with height
And m surfaces tilt since m = ug + fy
p
y
Baroclinic Atmosphere (temperature gradient)
x
This surface represents a surface where a parcel of air
Rising slantwise would be in equilibrium v  v shape
of surface depends on moisture distribution in environment
Weak shear
z
Strong shear
Absolute geostrophic
momentum surface
X
N
S
y
Consider a tube at X that is displaced to A
dw g
  v   v 
dt  v
dv
  f m  mg 
dt
At A, the tube’s v is less that its environment
At A, the tube’s m is greater than its environment
Tube will accelerate downward and southward…. Return to its original position
STABLE TO SLANTWISE DISPLACEMENT
This surface represents a surface where a parcel of air
Rising slantwise would be in equilibrium v  v shape
of surface depends on moisture distribution in environment
Weak shear
z
Strong shear
Absolute geostrophic
momentum surface
X
N
S
y
Consider a tube at B that is displaced to C
dw g
  v   v 
dt  v
dv
  f m  mg 
dt
At C, the tube’s v is greater that its environment
At A, the tube’s m is less than its environment
Tube will accelerate upward and northward…. Accelerate to D
UNSTABLE TO SLANTWISE DISPLACEMENT
Requirements for convection (slantwise or vertical)
Instability
Moisture
Lift
Evaluating Moist Symmetric Instability
Three different methods
1. Cross sectional analysis
1. Flow must be quasi-two dimensional on a scale of u0/f where u0 is the
speed of the upper level jet (e.g. 50 m s -1/10 -4 s -1 = 500 km)
2. Cross section must be normal to geostrophic shear vector (parallel to
mean isotherms) in the layer where the instability is suspected to be present
3. Air either must be saturated, or a lifting mechanism (e.g. ageostrophic
circulation associated with frontogenesis) must be present to bring the layer
to saturation.
4. Air must not be conditionally (or potentially) unstable, or inertially
unstable. If either condition is true, the vertical or horizontal instability will
dominate.
Two approaches depending on the nature of the lifting: is it
expected that a layer will be lifted to saturation or a parcel?
LAYER: Potential Symmetric Instability
PARCEL: Conditional Symmetric Instability
On cross section plot
On cross section
*
e
g
(superimpose on RHw or RHi to
determine saturation)
e vs M g
 vs M
(superimpose on RHw or RHi to
determine saturation)
Slantwise instability evaluation
Mg
M g  2M g
e
M g  2M g
Stable
z
Neutral
e  2e
z
z
Unstable
Mg
RH = 100%
y
e
RH = 100%
e  2e
red : e or 
e
y
*
e
blue: M g
e  2e
RH = 100%
y
green: RHw or RHi
Mg
Mg
 2 M g
Evaluating Moist Symmetric Instability
2. Evaluation of (saturation) equivalent geostrophic potential vorticity
Determining if CSI possible is equivalent to determining if the
saturation equivalent geostrophic potential vorticity is negative
MPVg ,s




 g   vg  fk  e*  0
u g  e* vg  e*  vg u g
  e*

 

 f 
0
p y
p x  x
y
 p
Determining if PSI possible is equivalent to determining if the
equivalent geostrophic potential vorticity is negative




MPVg  g   vg  fk  e  0
u g  e vg  e  vg u g
 

 

 f  e  0
p y p x  x
y
 p
Note that using the MPVg criteria does not differentiate between regions of CI/PI and CSI/PSI
An independent assessment of CI must be done to isolate regions of CSI/PSI
Evaluating Moist Symmetric Instability
3. Evaluation of slantwise convective available potential energy (SCAPE)
using single soundings
dv
  f m  mg 
dt
dw g
  v   v 
dt  v


Governing equations for displaced tube

g
PE   f m  m i      k  dl
2
g
1
v
v
Potential energy for reversible lifting of tube
v
Emanuel (1983, MWR, p.2018-19) shows that the maximum potential energy available to
a parcel ascending slantwise in an environment characterized by CSI occurs when the
parcel ascends along an Mg surface. SCAPE for this ascent is
2
SCAPE 
g

M ,1
 
 v   v k  dl
v
The susceptibility of the atmosphere to slantwise convection can be assessed by
reversibly lifting a (2-D) parcel along a surface of constant Mg and comparing its
virtual temperature (or v) to that of its environment
3 Dec 82
00 UTC
Surface
3 Dec 82
00 UTC
500 mb
3 Dec 82
12 UTC
Surface
3 Dec 82
12 UTC
500 mb
Meteorological conditions at the surface and 500 mb on 3 Dec 1982
Satellite images showing storm system – winter frontal squall line with trailing stratiform region
Cross section approximately normal to geostrophic shear showing
00Z during strong upright convection
Conditionally unstable: dominant mode
will be upright convection
e* and M g
12Z during more “stratiform” period
Neutral to slantwise convection : implies
that slantwise convective adjustment may
have occurred
Stable to upright
convection
Neutral to slantwise
convection
Dots take into
account
centrifugal
potential energy
To compare to
M surface
Moist Adiabat
for parcel lifted
from 690 mb
Sounding along
M = 40
T
Td
Neutral to slantwise
convection
Sounding along
M = 70
M = 70
M = 40
Nature of banding
Vertical velocity in model simulation: solid = upward, dashed = downward
Frontogenetic forcing in the
presence of small positive EPVg
Frontogenetic forcing in the
presence of large negative EPVg
As EPVg is reduced from positive values toward 0, the single updraft
becomes narrow and more intense. For more widespread and larger
negative EPVg the preferred mode becomes multiple bands