Section 8 Vertical Circulation at
Fronts
1. Structural and dynamical characteristics of
mid-latitude fronts
2. Frontogenesis
3. Semi-geostrophic equations
4. Symmetric instability
1. Structure and dynamical characteristics of mid-latitude fronts
EXAMPLES OF FRONTS
A front is a transition zone
between different air masses.
It is characterized by:
1. Larger than background
horizontal temperature
(density) contrasts ( strong
vertical shear)
2. Larger than background
relative vorticity
3. Larger than background
static stability
4. a quasi linear structure
(length >> width)
Let’s for the moment consider a zero-order front
We will assume that: 1) front is parallel to x axis
2) front is steady-state
3) pressure is continuous across the front
4) density and T are discontinuous across the front
dp 
p
y
dy 
p
z
dz
Warm side of front
dp w

 p 
 dy  
 dz
 z  w
w
p 
p 
dp c    dy    dz
z c
y c
Cold side of front
We have
 p
 
 y
dp c  dp w
and
p
z
 g

Substitute hydrostatic
equation and equate expressions:

  p
0   
   y

  p  
  
  dy    c   w  gdz
 c   y  w 
Solve for the slope of the front
dz
dy

 p 
 p 

  

 y  c  y  w
g  c   w 
For cold air to underlie warm air, slope must be positive
dz
dy

 p 
 p 

  

 y  c  y  w
g  c   w 
1) Across front pressure gradient on the cold side must be
larger that the pressure gradient on the warm side
Substituting geostrophic wind relationship u g  
dz
dy


f  wu g w   cu gc
1 p
f y

g  c   w 
u gw  u gc
2) Front must be characterized by positive geostrophic
relative vorticity du g
0
dy
The stronger the density (T) contrast becomes, the stronger is the vorticity at the front.
First-order fronts
1)
Larger than background horizontal temperature (density) gradient
2)
Larger than background relative vorticity
3)
Larger than background static stability
Working definition of a cold or warm front
The leading edge of a transitional zone that separates
advancing cold (warm) air from warm (cold) air, the length
of which is significantly greater than its width. The zone is
characterized by high static stability as well as larger-thanbackground temperature gradient and relative vorticity.
2. Frontogenetic Function
the Lagrangian rate of change of the magnitude of potential temperature gradient
Move to the whiteboard and talk
about 1D frontogenesis
3D Frontogenesis
F 
d

dt
Expanding the total derivative
d
dt


t
u

x
v

y
w

z
expanding the term involving the magnitude of the gradient

   
  
   
  
   
  
 
  z  
   x 
 y 
2
2
2
1/ 2
The solution
The Three-Dimensional Frontogenesis Function
d
F 

dt
becomes
F 

1





   1 p 0   dQ

   
x 
C p  p  x  dt

 u     v    w   
  
  

  
 x x   x y   x z 





   1 p 0   dQ
    

y 
C p  p   y  dt

 u    v    w   
  
  
  

 y x  y  y   y z 

(

 
  p 0


 z  C p

Compared to
      dQ     u      v      w    




p




 
 



  z 
dt     z  x    z  y    z  z  
 

F1D
)
d 
 d
u   v  
 

(
)
(
)


dt  x
x dt
x  x  x y
x p
• Confluence terms (or stretching
u
v
deformation): with x , y
• Shearing terms (or shearing deformation):
involved with
v
x
,
u
y
• Tilting terms: with derivative of omega

F 
1

(


   1  p 0    dQ
    
x 
C p  p  x  dt

 u     v    w   
  
  

  
 x x   x y   x z 


   1


y  C p

 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z  C p

      dQ     u      v      w    




p




 
 



  z 
dt     z  x    z  y    z  z  
 

)
dQ
The terms in the yellow box all contain the derivative
which is the diabatic heating rate. These terms are dt
called the diabatic terms.
F 
1

   1

x  C p

(
   1


y  C p


 p 0     dQ

 

 p    x  dt
    u      v      w    
  
  
  

    x  x    x  y    x  z  

 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z   C p

      dQ     u      v      w    


  
p
  
  

  z 
dt     z  x    z  y    z  z  
 

)


   1  p 0    dQ
    
F 
  x 
C p  p  x  dt
1






Horizontal gradient in
Temperature gradient
diabatic heating or cooling rate
 dQ 


If  x and x  dt  have the same sign, it means the diabatic

heating will increase the temperature gradient.

 / z 
  p 0
F 

    C p

      dQ   


p
 
  z 
dt   


3D
2D
D

3D

Vertical cross section of potential temperature
F 
1

   1

x  C p

(

 p 0     dQ

 

 p    x  dt
    u      v      w    
  
  
  

    x  x    x  y    x  z  

   1


y  C p

 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z  C p

      dQ     u      v      w    




p




 
 



  z 
dt     z  x    z  y    z  z  
 

The terms in this yellow box represent the contribution
to frontogenesis due to horizontal deformation flow.
)
F 
1

   1

x  C p

(
   1


y  C p


 p 0     dQ

 

 p    x  dt
u    v   
 
  

x x  x  y 
    u      v      w    
  
  
  

    x  x    x  y    x  z  

 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z   C p

      dQ     u      v      w    


  
p
  
  

  z 
dt     z  x    z  y    z  z  
 

)
Stretching
deformation

u    v   
 
  

y x  y  y 
Stretching Deformation


  / x  u    / y
F  




    x x 
Shearing
deformation
 v  


 
 y y 

Deformation
Deformation
acting on
acting on
temperature gradient temperature gradient
Stretching Deformation

  / x  u    / y
F  




    x x 
 v  


 
 y y 

Time = t + Dt
Time = t
y
y
T- 8DT
T- 7DT
T- 6DT
T- 5DT
T- 4DT
T- 3DT
T- 2DT
T- DT
T
x
T- 8DT
T- 7DT
T- 6DT
T- 5DT
T- 4DT
T- 3DT
T- 2DT
T- DT
T
x
F 
1

   1

x  C p

(
   1


y  C p


 p 0     dQ

 

 p    x  dt
u    v   
 
  

x x  x  y 
    u      v      w    
  
  
  

    x  x    x  y    x  z  

 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z   C p

      dQ     u      v      w    


  
p
  
  

  z 
dt     z  x    z  y    z  z  
 

)
Stretching
deformation

u    v   
 
  

y x  y  y 
Shearing Deformation


  / x
F  

 
 v    / y

 

 x y 
Shearing
deformation
 u  


 
 y x 

Deformation
Deformation
acting on
acting on
temperature gradient temperature gradient
Shearing Deformation

  / x
F  

 
 v    / y

 

 x y 
y
 u  


 
 y x 

y
x
x
F 
1

   1

x  C p

(

 p 0     dQ

 

 p    x  dt
    u      v      w    
  
  
  

    x  x    x  y    x  z  

   1


y  C p

 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z  C p

      dQ     u      v      w    




p




 
 



  z 
dt     z  x    z  y    z  z  
 

The terms in this yellow box represent the contribution
to frontogenesis due to tilting.
)
F 
1

   1

x  C p

(
   1


y  C p


 p 0     dQ

 

 p    x  dt
    u      v      w    
  
  
  

    x  x    x  y    x  z  

 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z   C p

      dQ     u      v      w    


  
p
  
  

  z 
dt     z  x    z  y    z  z  
 

)
Tilting terms

  / x  w    / y
F  




    x z 
Tilting
Of vertical
 Gradient
(E-W direction)
 w  


 
 y z 

Tilting
Of vertical
 Gradient
(N-S direction)
Weighting factor
Magnitude of gradient in one direction
Magnitude of total gradient
Tilting terms

  / x  w    / y
F  




    x z 
Before
z
 w  


 
 y z 

After
z
4D
4D
2D
2D


x or y
x or y
F 
1

   1

x  C p

(

 p 0     dQ

 

 p    x  dt
    u      v      w    
  
  
  

    x  x    x  y    x  z  

   1


y  C p

 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z  C p

      dQ     u      v      w    




p




 
 



  z 
dt     z  x    z  y    z  z  
 

The terms in this yellow box represent the contribution
to frontogenesis due to vertical shear acting on a
horizontal temperature gradient.
)
F 
1

   1

x  C p

(
   1


y  C p


 p 0     dQ

 

 p    x  dt
    u      v      w    
  
  
  

    x  x    x  y    x  z  

 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z   C p

      dQ     u      v      w    


  
p
  
  

  z 
dt     z  x    z  y    z  z  
 

)
Vertical shear acting on a horizontal temperature gradient
(also called vertical deformation term)
 / z   u    v   

 
F 
  

     z  x    z  y  
Vertical shear of E-W wind
Vertical shear of N-S wind
Component acting on
component acting on
a horizontal temp gradient in x a horizontal temp gradient in y
direction
direction
Vertical shear acting on a horizontal temperature gradient
 / z   u    v   

 
F 
  

     z  x    z  y  
Before

3D
6D
After
9D
6D
9D
3D

z
z
x
x
F 
1

   1

x  C p

(

 p 0     dQ

 

 p    x  dt
    u      v      w    
  
  
  

    x  x    x  y    x  z  

   1


y  C p

 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z  C p

      dQ     u      v      w    




p




 
 



  z 
dt     z  x    z  y    z  z  
 

)
The term in this yellow box represents the contribution
to frontogenesis due to divergence.

  / z  w  

F  





z

z





Compression
of vertical
 Gradient
by differential
vertical motion
Differential vertical motion

  / z  w  

F  





z

z





Before
z
After
z
4D
4D
2D

2D

x or y
x or y
2D Frontogenetic Function
F 
1


   1

x  C p

(
 p 0     dQ     u      v      w    

 
  

  
  

 p    x  dt     x  x    x  y    x  z  
   1


y  C p


 p 0     dQ     u      v      w    

 
  
  
 

   
 p    y  dt     y  x    y  y    y  z  

 
  p 0


 z  C p

      dQ     u      v      w    




p




 
 



  z 
dt     z  x    z  y    z  z  
 

)
The stretching and shearing deformations “look like” one another:
y
y
T- 8DT
T- 7DT
T- 6DT
T- 5DT
T- 4DT
T- 3DT
T- 2DT
T- DT
T
x
x
Another view of the 2D frontogenesis function
F2 D 
     u  
v    







     x    x  x  x  y    y
1
Recall the kinematic quantities:
D 
u
x

v
y

v
x
u

y
  u  
v  
 
 


y

x

y

y


divergence (D)
vorticity ()
stretching deformation (F1)
shearing deformation (F1).
F1 
u
x

v
y
F2 
v
x

u
y
and note that:
u
x

D  F1

2
v
x

  F2
u
2
y


F2  
2
v
y

D  F1
2
Substituting:
F2 D 
  
 
    x
1
   D  F1       F 2       

 


  
2
2
 
 x 
 y   y
   F2       D  F 1   
  



2
2
 x 
 y
 



F2 D 
  
 
    x
1
   D  F1       F 2       

 


  
2
2
 
 x 
 y   y
   F2       D  F 1   
  



2
2
 x 
 y
 



This expression can be reduced to:
F2 D
  

D
2      x
1
2
2
 
 
  F1 

 x
 y 

2
2
   
 
  2 F2 

  x  y  
 y 

 
y
y
x
x
Shearing and
stretching
deformation
“look alike” with
axes rotated
We can simplify the 2D
frontogenesis equation by
rotating our coordinate axes
to align with the axis of
dilatation of the flow (x´)
F2 D

1 
D  

2   


2
 2  2 



 F      

 x   y  



where F is the total deformation
F  (F1  F 2 )
2
2 1/ 2
F2 D

1 
D  

2   


2
 2  2 



 F      

 x   y  



This equation illustrates that horizontal frontogenesis is only
associated with divergence and deformation, but not vorticity

F2 D
Note that


  
F
D 

2 


F

2
2
 2  2 



     

 x   y  


 2  2 


       F cos 2    F cos 2 

 x   y  

F2 D 

2
( F cos 2   D )
Where
F is the total deformation of the

flow, β is the angle between the
isentropes and the dilatation axis of the

total deformation field, and D is
divergence (D <0 for convergence)
F2 D 

2
( F cos 2   D )
Frontogenesis occurs
1)if non-zero  is coincident with convergence (D<0)
2)if the total deformation field (F) acts on isentropes that are between

0 and 45° of the dilatation axis of the total deformation. deformation.

y
y
x
x
3. S.G. vs. Q.G. Approximations
u  ug  ua ,
v  vg  va ,
f  const
du

 fv 

 dt
x

dv   fu  

y
 dt
• Q.G.:
S.G.
u g  u a ,
v g  v a

d g u g
 fv a

 dt

d g v g
  fu a

 dt
dg
where
dt


t
 ug

x
 vg
u  ug ,
du g
dt

y
 fv a
v  vg  va
Sawyer-Eliassen Equation
    2  M   2
 M  2
 d  
 
 
 
 2  2

 2  Q g  
 dt 
p
y
p
py
y
p

y







Geostrophic deformation
Diabatic heating
Right side of equation represent the forcing
(known from measurements or in model solution)
, the streamfunction, is the response.
V and ω can be derived from 
Questions: 1) How is the thermal wind balance maintained by the transverse circ.?
2) Where should we expect upward motion (precipitation)?
du g
dt
 fv a

Cold
warm
warm
cold
Nature of the solution of the Sawyer-Eliassen Equation:
A direct circulation (warm air rising and cold air
sinking) will result with positive forcing.
An indirect circulation (warm air sinking and cold air
rising) will result with negative forcing.
Warm air
Cold air
 d  

   0
(heating in the warm side
y  dt 
and cooling in the cold side) will produce
A thermally direct circulation and promote
Frontogenesis.

Cold air
Warm air
 U g  V g  

Q g  2  



y

x

y

y


Geostrophic
shearing
deformation
Geostrophic
stretching
deformation
Qg
ST
 2
V g 
Geostrophic stretching deformation
y y
Entrance region of jet
Note in this figure that both
V g
y
and

y
are negative, implying
frontogenesis and a direct circulation in which warm air is rising and
cold air sinking.
Qg
sH
 2
U g 
y
Geostrophic shearing deformation
x
confluent flow
along front
Note in this figure that both
U
g
and

are positive, implying
x
y
frontogenesis and a direct circulation in which warm air is rising and
cold air sinking.
Why does a spinning top stay
upright?
• Buoyancy tends to stabilize air parcels
against vertical displacements, and rotation
tends to stabilize parcels with respect to
horizontal displacements.
• If ordinary static and inertial stabilities are
satisfied, is the flow always stable?
4. Symmetric Instability
• hydrostatic instability
d v
0
stable
dz
d v
0
neutral
dz
d v
0
unstable
dz
• Inertial instability

u g 
f  f 
  0 stable
y 



u g 
f  f 
  0 neutral
y 



u g 
f  f 
  0
y 

unstable
MSI: an intuitive explanation
M = absolute zonal momentum
30
40
M = fy-ug
dM/dy>0
60
70
see also: Jim Moore’s meted module on frontogenetic circulations & stability)
Potential
Potential Symmetric INstability
Potential Symmetric Stability
-
Dash: e
Solid: Mg
-
-
-
-
-
-
Symmetric instability evaluation
The flow satisfies
M
g
M
g
 2DM

u g 
 0 and f  f 
  0
y

z
dz
d v
M
g
Stable

y
e
g
e
g
z
M
Neutral
 e  2D e
z
z
Unstable
g
 e  2D e
red :  e or  v

 2DM
y
e
 e  2D e
y
M
g
M
g
 2DM
blue : M
g
g