The Semi-geostrophic system of equations
For development of these equations, we will use “geostrophic coordinates”
where the x axis is along the front (the isentropes) and the y axis is positive
toward the cold air.
We will assume that the front is 2D – no along front gradients
The geostrophic wind relationships are given by
Ug  
1 
f y
1 
Vg 
f x
The hydrostatic relationship is given by
1 
 
f p
R  p0 
 

fp0  p 
where
And the thermal wind relationships are given by
U g
p


y
Vg
p
 

x
cv / c p
The equation of motion in the along front direction
dU g
dt

du
 fv
dt
Where Ug and u are the along front geostrophic
and ageostrophic wind components
d
And the along front geopotential height gradient dx
is zero because of the 2-D assumption
We will now make the “geostrophic momentum approximation”
Assume that there is no systematic change in the along-front ageostrophic wind
dU g
dt

du
dt
The equation of motion in the along front direction becomes
dU g
dt
 fv
dU g
dt
 fv
Expand the total derivative:
dUg
dt

U g
t
Ug
U g
x
u
U g
x
 Vg
U g
y
v
U g
y

U g
p
 fv
The thermodynamic equation:
d 






Ug
u
 Vg
v

dt t
x
x
y
y
p
Since the along front flow is assumed to be in geostrophic balance
we will ignore the crossed out terms
dUg
dt

U g
t
Ug
U g
x
 Vg
U g
y
v
U g
y

U g
p
 fv
We will now introduce the absolute geostrophic momentum
M  U g  fy
Note that M is conserved:
U g
t
Ug
U g
x
dM dU g
dy dU g

f

 fv  0
dt
dt
dt
dt
 Vg
U g
M
M
v

0
y
y
p
U g
t
Ug
U g
x
 Vg
U g
y
v
M
M

0
y
p
d 





Ug
 Vg
v

dt t
x
y
y
p
A
B


Take
of A and add it to  
of B

y
p
  
    M
M
  v
     v

y  y
p  p  y
p
 U g U g Vg U g 

  d 
    
  2

p y 
y  dt 

 p x
Now consider the continuity equation
u v 
 
0
x y p
Define a streamfunction for the ageostrophic flow in the y direction

v
p


y
Substitute into the equation below
  
    M
M
  v
     v

y  y
p  p  y
p
 U g U g Vg U g 

  d 
    
  2

p y 
y  dt 

 p x
to get

   2  M   2  M   2
  d 
  
 2   2

 2  Qg    
  
p  y
y  dt 

 p  py  y  p
Where:
 U g Vg Vg U g 

Qg  2

y p 
 y p
is the geostrophic forcing function

   2  M   2  M   2
  d 
  
 2   2

 2  Qg    
  
p  y
y  dt 

 p  py  y  p
This equation is called the “Sawyer-Eliassen circulation equation” derived in
papers by Sawyer (1955) and Eliassen (1962)
It has the form:
 2u
 2u
 2u
u
u
A 2 B
 C 2  D  E  Fu  G
x
xy
y
x
y
Which has a solution that depends on the sign of the discriminant
B 2  4 AC  0
Elliptic
B 2  4 AC  0
Parabolic
B 2  4 AC  0
Hyperbolic
B 2  4 AC
Elliptic solutions to the equation
 2u
 2u
 2u
u
u
A 2 B
 C 2  D  E  Fu  G
x
xy
y
x
y
are those in which u is uniquely determined from the forcing G
We therefore require
 M
 2
 p
4
2


  M
  4  
 
p  y



  0

B 2  4 AC  0
M



Use thermal wind relationship
p
y
M 
 M
 4
0
p y
p y
  M  M 
  0

 p y y p 
 
This is the QG potential vorticity!
The Sawyer-Eliassen equation will have solutions provided that the air
is statically stable and inertially stable, that is, the QG potential vorticity
is positive. If QG Potential Vorticity is negative, solutions are non-unique
because of the release of the instability.
Static stability
Baroclinicity (thermal wind)
Inertial stability

   2  M   2  M   2
  d 
  
 2   2

 2  Qg    
  
p  y
y  dt 

 p  py  y  p
Geostrophic deformation
Diabatic heating
Right side of equation represent the forcing
(known from measurements or in model solution)
, the streamfunction, is the response
Solutions for  can be obtained provided lateral and top/bottom boundary
conditions are specified and the potential vorticity is positive in the domain
(air is inertially, convectively and symmetrically [slantwise] stable).
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.
Cold air
Warm air
Cold air
Warm air
Dynamics of frontogenesis
A conceptual model of the ageostrophic circulation caused by frontogenesis
On the figure on the left,
Dashed lines: potential temperature
Blue lines: pressure surfaces (exaggerated)
Shading: isotachs (blue into screen, red out)
1. Initial condition
Geostrophically-balanced
weak front
2. Impulsively intensify front
Stronger temperature gradient
leads to more steeply sloped
pressure surfaces and an increase
in the pressure gradient force
at both high and low levels
Dynamics of frontogenesis
A conceptual model of the ageostrophic circulation caused by frontogenesis
2. Impulsively intensify front
Stronger temperature gradient
leads to more steeply sloped
pressure surfaces and an increase
in the pressure gradient force
at both high and low levels
3. Air accelerates
Air rises on warm side
Air descends on cold side
Air accelerates along isentropes
toward cold air and into screen aloft
Air accelerates toward warm air and
out of screen in low levels
Dynamics of frontogenesis
A conceptual model of the ageostrophic circulation caused by frontogenesis
3. Air accelerates
Air rises on warm side
Air descends on cold side
Air accelerates toward cold air and
into screen aloft
Air accelerates toward warm air and
out of screen in low levels
4. Balance is restored
- Air rises and cools on warm side
- Air sinks and warms on cold side
- counteracts effects of frontogenesis
Air cools at moist
Adiabatic lapse rate
Air warms at dry
Adiabatic lapse rate
-Wind speed in upper jet increases
(into screen)
-Wind speed in lower jet increases
(out of screen)
- Coriolis force increases
- Geostrophic balance restored
The circulation describe in the last few slides can be seen clearly on the front
illustrated on the cross section below
An example of a solution to the SE Equation (left) along cross section AA’
in the cyclone illustrated on the right (from Han et al. 2007).
Note the circulation along the front (dashed lines are potential temperature)
and within the trowal region (heavy dashed line) with sinking motion in the dry
air to the south of the trowal.
An analysis of the SE forcing term
 U g Vg Vg U g 

Qg  2


y

p

y

p


Let’s rewrite using the thermal wind relationships
 U g  Vg  

Qg  2 

 y x y y 
Geostrophic
shearing
deformation
Geostrophic
stretching
deformation
We will examine each of these terms in isolation
Using the expression for the nondivergence of the geostrophic wind
u g
x

This expression can be written:
vg
y
 u g  u g  

 2 

 x y y x 

=0
x
In the entrance quadrant ug increases with x
while  decreases with y.
Consider a jetstreak where
 u g  u g  
  0
 2 

 x y y x 
DIRECT
CIRCULATION
In the exit quadrant ug decreases with x
while  decreases with y.
 u g  u g  
  0
 2 

 x y y x 
INDIRECT
CIRCULATION
0
 u g  u g  

 2

 x y y x 
Consider a shear zone along a temperature
gradient where u g =0
x
ug decreases with y while  increases with x.
 u g  u g  
  0
 2

 x y y x 
INDIRECT
CIRCULATION
Cold advection pattern corresponds to an indirect circulation
Correspondingly:
Warm advection pattern corresponds to an direct circulation
Example of solution of the
Sawyer-Eliassen equation
The circulation about an
ideal frontal zone
characterized by
Confluence (top)
Shear (bottom)
Streamlines of ageostrophic circulation (thick solid lines)
Isotachs of ug (denoted U) (dashed lines)
Isotachs fo vg (denoted V) (thin solid lines)
A second example:
Qg sH
U g 
 2
y x
Geostrophic shearing deformation
confluent flow
along front
Note in this figure that both
U g
and

x
are positive, implying
y
frontogenesis and a direct circulation in which warm air is rising and
cold air sinking.
Qg ST  2
Vg 
y y
Geostrophic stretching deformation
Entrance region of jet
Note in this figure that both
Vg
and

y
are negative, implying
y
frontogenesis and a direct circulation in which warm air is rising and
cold air sinking.
Upper level fronts and frontogenesis
The essential mechanism for formation
of upper level fronts
An indirect circulation tilts the isentropes
associated with the stable
stratospheric air and vortex tubes
associated with vertical shear into
the vertical creating:
1) A strong thermal gradient
2) Enhanced vertical vorticity
3) A region of high static stabilty
THE ESSENTIAL DYNAMICAL
CHARACTERISTICS OF A FRONT
Also
High
Ozone
Measurements of Potential Vorticity and Strontium 90 in 1963 across an upper level front
Pot. Temp and wind speed
Potential Vorticity
Cyclonic Shear Boundary
Tropopause
And fronts
Absolute Angular Momentum
Absolute Angular Momentum/Pot Temp
Extremely high
resolution measurements
of upper level frontal
structure made with a
research aircraft
supplemented by sondes
(Shapiro 1981)
Upper level frontogenesis
Consider first a jetstreak:
Shearing deformation
Qg ST  2
Vg 
U g 
 2
y y
x y
forces frontolytic vertical circulation in exit region leading
to tilting of the isentropes in a manner that supports upper level
frontogenesis!
Consider next the shearing term Qg sH
U g 
 2
y x
Cold advection in the presence of cyclonic shear produces an indirect circulation
leading to descending motion in warm air and ascending motion in cold air
which tilts the isentropes, leading to upper level frontogenesis!
anticyclonic shear
cold advection
cyclonic shear
cold advection
direct
indirect
cyclonic shear
warm advection
direct
anticyclonic shear
warm advection
indirect
Effect of temperature advection on the
Vertical circulation about a straight jetstreak
Straight jetstreak: no temperature advection
Straight jetstreak: cold advection
THIS PATTERN WILL CAUSE
DOWNWARD MOTION ALONG JET
AXIS, LEADING TO RAPID UPPER
LEVEL FRONTOGENESIS
Straight jetstreak: warm advection
Evolution of 500 mb heights,
temperature and absolute
vorticity over a 48 hr period (12
hour time intervals)
As a jetstreak propagates down
the back side of trough in region
of cold advection, look at the
rapid change in vorticity in the
last 24 hours
Cyclonic vorticity created by upper
level frontogenesis!