### 11. Frontogenesis

```Structure and dynamical characteristics of mid-latitude fronts
Front: A boundary whose primary structural and dynamical
characteristic is a larger then background density (temperature) contrast
A zero-order front: A front characterized by a discontinuity in temperature and density
This type of front does not occur in the atmosphere, but does exist where two
fluids of different density approach one another as illustrated below
mixing associated with friction prevents atmospheric fronts from becoming zero-order
ATMOSPHERIC FRONTS
Gradients in temperature and density are discontinuous across fronts
Let’s for the moment consider a zero-order front
We will assume that: 1) pressure must be continuous across the front
2) front is parallel to x axis
p
p
dp 
dy  dz
y
z
Warm side of front
 p 
 p 
dpw    dy    dz
 z  w
 y  w
Cold side of front
 p 
 p 
dpc    dy    dz
 z c
 y c
Substitute hydrostatic equation and equate expressions:
 p   p  
0        dy   c   w gdz
 y  c  y  w 
Solve for the slope of the front
 p   p 
    
dz  y c  y  w

dy
g  c   w 
For cold air to underlie warm air, slope must be positive
Therefore:
 p   p 
    
dz  y c  y  w

dy
g  c   w 
1) Across front pressure gradient on the cold side must be
larger that the pressure gradient on the warm side
1 p
Substituting geostrophic wind relationship u g  
f y

dz f  wu g w  cu gc

dy
g  c   w 

ug w  ugc
2) Front must be characterized by positive geostrophic
relative vorticity dug
dy
0
Real (first order) fronts
1)
Larger than background horizontal temperature (density) contrasts
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 gradients in temperature and relative vorticity.
EXAMPLES OF FRONTS
EXAMPLES OF FRONTS
EXAMPLES OF FRONTS
Frontogenesis
Ageostrophic Circulations associated with fronts and jetstreaks
The formation of a front is called frontogenesis
The decay of a front is called frontolysis
These processes are described quantitatively in terms of the
Three-Dimensional Frontogenesis Function
d
F  
dt
Where
 is the magnitude of the 3-D potential temperature gradient
d
and the total derivative
dt
implies that the change in the  gradient is calculated
following air-parcel motion
The processes by which a front forms or decays can be understood
more directly by expanding the frontogenetical function
d
F  
dt
Algebraically, this involves expanding the total derivative
d




 u v  w
dt t
x
y
z
expanding the term involving the magnitude of the gradient
1/ 2
         
          
 x   y   z  
2
2
2
Reversing the order of differentiation, differentiating, and
k
d  p0  1 dQ
  
then using the thermodynamic equation
dt  p  C p dt
d
to replace the term
in the resulting equation.
dt
The solution
The Three-Dimensional Frontogenesis Function
d
F  
dt
becomes


1   1  p0     dQ   u    v    w  





F



 
 


    

 x  C p  p   x  dt   x x   x y   x z 




  1  p0     dQ   u    v    w  

















y  C p  p   y  dt   y x   y y   y z 



 
 p0      dQ   u    v    w  

  

p
  
  




z 
dt   z x   z y   z z 
 C p   z 

(
)


1   1  p0     dQ   u    v    w  





F



 
 


    

 x  C p  p   x  dt   x x   x y   x z 




  1  p0     dQ   u    v    w  















   








y  C p  p   y  dt   y x   y y   y z 



 
 p0      dQ   u    v    w  




p











 z x  z y 
  z 
z 
C
dt

z

z
 
 

 
 p 

(
)
dQ
The terms in the yellow box all contain the derivative
dt
which is the diabatic heating rate. These terms are
called the diabatic terms.


1   1  p0     dQ   u    v    w  





F



 
 


    

 x  C p  p   x  dt   x x   x y   x z 




  1  p0     dQ   u    v    w  















   








y  C p  p   y  dt   y x   y y   y z 



 
 p0      dQ   u    v    w  




p











 z x  z y 
  z 
z 
C
dt

z

z
 
 

 
 p 

(
)
The terms in this yellow box represent the contribution
to frontogenesis due to horizontal deformation flow.


1   1  p0     dQ   u    v    w  





F



 
 


    

 x  C p  p   x  dt   x x   x y   x z 




  1  p0     dQ   u    v    w  















   








y  C p  p   y  dt   y x   y y   y z 



 
 p0      dQ   u    v    w  




p











 z x  z y 
  z 
z 
C
dt

z

z
 
 

 
 p 

(
)
The terms in this yellow box represent the contribution
to frontogenesis due to vertical shear acting on a


1   1  p0     dQ   u    v    w  





F



 
 


    

 x  C p  p   x  dt   x x   x y   x z 




  1  p0     dQ   u    v    w  















   








y  C p  p   y  dt   y x   y y   y z 



 
 p0      dQ   u    v    w  




p











 z x  z y 
  z 
z 
C
dt

z

z
 
 

 
 p 

(
)
The terms in this yellow box represent the contribution
to frontogenesis due to tilting.


1   1  p0     dQ   u    v    w  





F



 
 


    

 x  C p  p   x  dt   x x   x y   x z 




  1  p0     dQ   u    v    w  















   








y  C p  p   y  dt   y x   y y   y z 



 
 p0      dQ   u    v    w  




p











 z x  z y 
  z 
z 
C
dt

z

z
 
 

 
 p 

(
)
The term in this yellow box represents the contribution
to frontogenesis due to divergence.
1
F


x
(


 1  p0     dQ   u    v    w  

  
  
  

    
C
p
x dt   x x   x y   x z 

 p   



 1  p0     dQ   u    v    w  

  
  

  
    
C
p
y dt   y x   y y   y z 

 p   


 
 p0      dQ   u    v    w  

  

p
  
  




z 
dt   z x   z y   z z 
 C p   z 



y
)


 / x  1  p0     dQ  



F
 
    
  C p  p   x  dt  


for specific
heat of air and
air pressure
Weighting
factor
diabatic heating or cooling rate
Magnitude of  gradient in one direction


 / x  1  p0     dQ  


F
 
    
in x direction
  C p  p   x  dt  




 / y  1  p0     dQ  
F
 
    
in y direction
  C p  p   y  dt  


Can you think of other examples where this term might be important to frontogenesis?
1
F


x
(


 1  p0     dQ   u    v    w  

  
  
  

    
C
p
x dt   x x   x y   x z 

 p   



 1  p0     dQ   u    v    w  

  
  

  
    
C
p
y dt   y x   y y   y z 

 p   


 
 p0      dQ   u    v    w  

  

p
  
  




z 
dt   z x   z y   z z 
 C p   z 



y
)


 p0      dQ  

p

 



dt  

 C p   z 

 / z
F

for specific
heat of air
Weighting
factor
diabatic heating or cooling rate
Magnitude of  gradient in one direction
 / z
F

3D
2D
D

3D



 p0      dQ  

p
 



dt  

 C p   z 

1
F


x
(


 1  p0     dQ   u    v    w  

  
  
  

    
C
p
x dt   x x   x y   x z 

 p   



 1  p0     dQ   u    v    w  

  
  

  
    
C
p
y dt   y x   y y   y z 

 p   


 
 p0      dQ   u    v    w  

  

p
  
  




z 
dt   z x   z y   z z 
 C p   z 



y
 u    v  


  
 x x   x y 
Stretching
deformation
)
 u    v  

  

 y x   y y 
Stretching Deformation

  / x  u    / y  v  



F  


  y y 

   x x 

Deformation
Deformation
acting on
acting on
Weighting factors
Magnitude of  gradient in one direction
Shearing
deformation
Stretching Deformation

  / x  u    / y  v  



F  


  y y 

   x x 

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
1
F


x
(


 1  p0     dQ   u    v    w  

  
  
  

    
C
p
x dt   x x   x y   x z 

 p   



 1  p0     dQ   u    v    w  

  
  

  
    
C
p
y dt   y x   y y   y z 

 p   


 
 p0      dQ   u    v    w  

  

p
  
  




z 
dt   z x   z y   z z 
 C p   z 



y
 u    v  


  
 x x   x y 
Stretching
deformation
)
 u    v  

  

 y x   y y 
Shearing Deformation

  / x  v    / y  u  


 


F  
  y x 

   x y 

Deformation
Deformation
acting on
acting on
Weighting factors
Magnitude of  gradient in one direction
Shearing
deformation
Shearing Deformation

  / x  v    / y  u  


 


F  
  y x 

   x y 

y
y
x
x
1
F


x
(


 1  p0     dQ   u    v    w  

  
  
  

    
C
p
x dt   x x   x y   x z 

 p   



 1  p0     dQ   u    v    w  

  
  

  
    
C
p
y dt   y x   y y   y z 

 p   


 
 p0      dQ   u    v    w  

  

p
  
  




z 
dt   z x   z y   z z 
 C p   z 



y
)
Vertical shear acting on a horizontal temperature gradient
(also called vertical deformation term)
F 
 / z

  u    v   


  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
Weighting factor
Magnitude of  gradient in one direction
Vertical shear acting on a horizontal temperature gradient
 / z
F 

  u    v   


  z x    z y  



Before

3D
6D
After
9D
6D
9D
3D

z
z
x
x
1
F


x
(


 1  p0     dQ   u    v    w  

  
  
  

    
C
p
x dt   x x   x y   x z 

 p   



 1  p0     dQ   u    v    w  

  
  

  
    
C
p
y dt   y x   y y   y z 

 p   


 
 p0      dQ   u    v    w  

  

p
  
  




z 
dt   z x   z y   z z 
 C p   z 



y
)
Tilting terms

  / x  w    / y  w  



F  


  y z 

   x z 

Tilting
Of vertical
(E-W direction)
Tilting
Of vertical
(N-S direction)
Weighting factor
Magnitude of  gradient in one direction
Tilting terms

  / x  w    / y  w  



F  


  y z 

   x z 

Before
z
After
z
4D
4D
2D
2D


x or y
x or y
1
F


x
(


 1  p0     dQ   u    v    w  

  
  
  

    
C
p
x dt   x x   x y   x z 

 p   



 1  p0     dQ   u    v    w  

  
  

  
    
C
p
y dt   y x   y y   y z 

 p   


 
 p0      dQ   u    v    w  

  

p
  
  




z 
dt   z x   z y   z z 
 C p   z 



y
)
Differential vertical motion
(also called divergence term because
w/ z is related to divergence through continuity equation)

  / z
F  

 

 w  



z

z



Compression
of vertical
by differential
vertical motion
Weighting factor
Magnitude of  gradient in one direction
Differential vertical motion

  / z
F  

 

 w  


 z z 

Before
z
After
z
4D
4D
2D

2D

x or y
x or y
Another view of the 2D frontogenesis function
F2 D 
1

   u  v      u  v  
   




 
 x  x x x y   y  y x y y 
Recall the kinematic quantities:
D
u v

x y
 
v u

x y
divergence (D)
vorticity ()
stretching deformation (F1)
shearing deformation (F1).
F1 
u v

x y
F2 
v u

x y
and note that:
u D  F1

x
2
v   F2

x
2
u F2  

y
2
v D  F1

y
2
Substituting:
F2 D
1     D  F1      F2       F2      D  F 1    




    


  

   x  2  x  2  y   y  2  x  2  y  
F2 D
1     D  F1      F2       F2      D  F 1    




    


  

   x  2  x  2  y   y  2  x  2  y  
This expression can be reduced to:
F2 D
1

2 
   2  2 
  2  2 
   




D


F


2
F
 
2
 x y 
 1  x


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 dilitation
of the flow (x´)
F2 D
2
2


1 
2

       

  

D   F1    
2  
x   y   







F2 D
2
2


1 
2

       





D   F1    

2  
x   y   







This equation illustrates that horizontal frontogenesis is only
associated with divergence and deformation, but not vorticity
Yet another view of the 2D frontogenesis function
Let’s replace u and v with their geostrophic components and
examine geostrophic frontogenesis:
F2 Dg
1


   u g  vg      u g  vg  
   




 
 x  x x x y   y  y x y y 
Recalling the Q vector
1
Q1
f
1
Q2
f


 Vg

  Vg
 
ˆ



Q  f  
  i , 
   ˆj 
  y
 
 x
 
 
Therefore:
F2 Dg

 1  Q  
  
 f  
Magnitude of geostrophic
frontogenesis is a scalar multiple
of the cross isentropic
component of the Q vector
Convergence of Q vectors
associated with rising motion
Divergence of Q vectors
associated with descending
motion
Implication: Direct circulation
(warm air rising and cold air
sinking) associated with
frontogenesis
Is geostrophic frontogenesis, as represented by the Q vector,
sufficient to describe the circulation about a front?
Consider a simple north-south front undergoing frontogenesis by the geostrophic wind
Assume that the confluence occurs at a constant rate k
u g 
d  d 

k


dt  dx 
x x
x

 d 
d

k
dt

x
 dx 
integrate to get:
      kt
    e
 x t  x 0
Using typical values of
u g
x
it takes 105 seconds or about 1 day for geostrophic
confluence to increase the temperature gradient by a factor of e (2.5)
example from real atmosphere
In 6 hours, temperature
gradient doubles, a factor of 8
larger than that expected from
scale analysis of geostrophic
confluence
Implication: ageostrophic non-QG
forcing is important to the
circulations on cross frontal scale
QUASI-GEOSTROPHIC THEORY IS
INSUFFICIENT TO ACCOUNT FOR
THE VERTICAL MOTIONS IN THE
VICINITY OF FRONTS
```