THE TRENBERTH (1978) INTERPRETATION
The Quasi-Geostrophic Omega Equation (without friction and diabatic terms)



 2
  
2 
 
2
  f 0 2   f 0
Vg   g  f    Vg    
p 
p
 p 


PROBLEM: TERM 1 and 2 on the RHS are often large and opposite
leading to ambiguity about the sign and magnitude of  when
analyzing weather maps
Trenberth (1978) argued that carrying out all of the derivatives on the RHS on the
 Equation could simplify the forcing function for .
We will now develop the Trenberth (1978)* modification to the
QG Omega equation
*Trenberth, K.E., 1978: On the Interpretation of the Diagnostic Quasi-Geostrophic Omega Equation. Mon. Wea. Rev.,
106, 131–137
QG OMEGA EQUATION:



 2
  
2 
 
2
  f 0 2   f 0
Vg   g  f    Vg    
p 
p
 p 


EXPAND THE ADVECTION TERMS:
 g  f 
 2 f0  2 
   g  f 
 2
 2 
2
  f 0
   
 vg
 vg

u g
    u g
2 


p

p

x

y

x

p

y

p






USE THE EXPRESSIONS FOR THE GEOSTROPHIC WIND AND
GEOSTROPHIC VORTICITY:
ug  
1 
f y
vg 
1 
f x
g  
1 2

f0
To Get:
 2 f0  2 
  1    1 2
   

   


f

0
2 


p

p
f

y

x


 f0

 1    1 2
   
f  
f

x

y

 f0
 1 2    2   2 
f    


f

y

x

p
x yp 


 2 f0  2 
  1    1 2
   

   


f

0
2 


p

p
f

y

x


 f0

 1    1 2
   
f  
f

x

y

 f0
 1 2    2   2 
f    


f

y

x

p
x yp 


EXPAND ALL THE DERIVATIVES
THEN USE THE JACOBIAN OPERATOR TO SIMPLIFY NOTATION
 A B A B 

J  A, B   

 x y y x 
RESULT:
 2 f0  2 
1

F1  F2 
   


2 
 p 
f

    2 
   2 
 2   
2  
  J  ,

  2 J  ,
F1   J    ,   J   , 
p  
p    x xp 

 y yp 
 
  
 
 

F2  J  ,  2   J  , ff 0   J  ,  2
p 
 p
  p
 
    2 
   2 
 2   
2  
  J  ,

  2 J  ,
F1   J    ,   J   , 

p

p

x

x

p

y

y

p

 
  



Opposite term:
opposite sign = same term
Same term:
opposite sign
 
  
 
 

F2  J  ,  2   J  , ff 0   J  ,  2
p 
 p
  p
 
Deformation
Terms in Sutcliff eqn
Trenberth: Ignore deformation terms (removes frontogenetic effects)
 2 f0  2 
1

F1  F2 
   


2 


p
f




1
F1  F2   1 2 J   ,  2   J   , ff 0 
f
f   p
  p

Approximate last term = 2  last term


1
F1  F2   2  J   ,  2   J   , ff 0 
f
f   p
  p



1
F1  F2   2  J   ,  2   J   , ff 0 
f
f   p
  p

Expand Jacobian terms:



Vg
 2 f0  
  2 f 0
   
  g  f 
2 
 p 


 p
2
This result says that large scale vertical motions can be diagnosed by
Examining the advection of absolute vorticity by the thermal wind
RECALL SUTCLIFF’S EQUATION





f 0 V  V    V0  V    g 0   g  f 
SAME INTERPRETATION!!
The Geostrophic paradox
Confluent geostrophic
flow will tighten
temperature gradient,
leading to an increase in
shear via the thermal wind
relationship……..
……but advection of
geostrophic momentum
by geostrophic wind
decreases the vertical
shear in the column
so…geostrophic flow destroys
geostrophic balance!
The geostrophic paradox: a mathematical interpretation
 

  Vg   vg  f 0uag  0
 t

y momentum equation (QG)
 
  
  Vg         0
 t
 p 
Thermodynamic energy equation (QG)
For the moment, let’s ignore the ageostrophy (no uag and no )
 

  Vg   vg  0
 t

Let’s look at this equation
 
  

V



    0
g
 t
 p 
Take vertical derivative of first equation
vg
vg 
   
  vg
 
  0
f0
 ug
 vg
  Vg   vg   f 0 

p  t
p  t
x
y 
 
vg
vg 
   
  vg
 
  0
f0
 ug
 vg
  Vg   vg   f 0 

p  t
p  t
x
y 
 
Expand the derivative:
 u g vg vg vg 
   
   
 vg
f0
 f0 

  Vg   vg     Vg    f 0
0

p  t
   t
 p
 p x p y 
Substitute using the thermal wind relationship:
vg
 2
f0

p xp
u g
 2
f0

p yp
to get:

  
   
   
 vg   Vg
  
f0
     0
  Vg   vg     Vg    f 0

p  t
   t
 p   x
 p 
Remember equation in blue box
The geostrophic paradox: a mathematical interpretation
 

  Vg   vg  f 0uag  0
 t

y momentum equation (QG)
 
  
  Vg         0
 t
 p 
Thermodynamic energy equation (QG)
For the moment, let’s ignore the ageostrophy (no uag and no )
 

  Vg   vg  0
 t

 
  

V



    0
g
 t
 p 
Now let’s look at this equation
Take x derivative of second equation
  
     
   
   
  
   Vg             u g     vg     0
x  t
x  t  p 
x  p 
y  p 
 p 
  
     
   
   
  
   Vg             u g     vg     0
x  t
x  t  p 
x  p 
y  p 
 p 
Expand the derivative and use vector notation:

2



  
 
    
    Vg
  
   Vg         Vg   
     0
x  t
 p   t
 xp   x
 p 
Now recall first saved equation:
   
   
 vg
f0
  Vg   vg     Vg    f 0

p  t
   t
 p


 Vg
  
  
     0
 p 
  x
Let’s take these two blue boxed equations and compare them…..
2
 
   
 
  Vg    
 t
  xp 

 Vg
  
   

 p 
 x

 Vg
  
 
  vg 
   
   
  Vg     f 0
 t
  p 
 p 
 x
vg
 2
f0

p xp
thermal wind
balance
Following the geostrophic wind the
magnitude of the temperature gradient
and the vertical shear have opposite
Tendencies
TIGHTENING THE TEMPERATURE
GRADIENT WILL REDUCE THE SHEAR!
The Geostrophic paradox:
RESOLUTION
A separate “ageostrophic
circulation” must exist that
restores geostrophic balance that
simultaneously:
1) Decreases the magnitude of
the horizontal temperature
gradient
2) Increases the vertical shear
The Q-Vector interpretation of the Q-G Omega Equation
(Hoskins et al. 1978)

2
 Vg
  
 
   
  
   
  Vg    
From consideration of the

t

x

p

x


 p 


geostrophic wind, we derived

these equations:


Vg
  

  vg 
   
   
  Vg     f 0
 t
  p 
 p 
 x
Let’s denote the term on the RHS:

Vg
  
Q1  
   
x
 p 
If we start with our original equations, below,
 

  Vg   vg  f 0uag  0
 t

y momentum equation (QG)
 
  
  Vg         0
 t
 p 
Thermodynamic energy equation (QG)
and perform the same operations as before, but with the ageostrophic terms
included….
We arrive at:
 
 vg
  Vg    f 0
 t
 p
u

  Q1  f 02 ag  0
p

With ageostrophic terms
2

 
   
  Q1  
0
  Vg   
x
 t
 xp 
 
  v g 
  Q1  0
  Vg     f 0
 t
  p 
Only geostrophic terms
2
 
   
  Q1  0
  Vg    
 t
  xp 
Note that the additional terms represent the ageostrophic circulation that
works to reestablish geostrophic balance as air accelerates in unbalanced flow.
 
 vg
  Vg    f 0
 t
 p
u

  Q1  f 02 ag  0
p

With ageostrophic terms
2

 
   
  Q1  
0
  Vg   
x
 t
 xp 
Let’s multiply the bottom equation by -1 and add it to the top equation, recalling
that
u g
 2
f0

p yp

2 uag
 2Q1  
 f0
x
p
Let’s do the same operations with the x equation of motion and the thermodynamic
equation. If we do, we find that:

2 vag
 2Q2  
 f0
y
p
Let’s do the same operations with the x equation of motion and the thermodynamic
equation.
 
 u g
  Vg    f 0
 t
 p
vag

2
  Q2  f 0
0
p

2

 
   



V



Q


0


g
2

y
 t
 xp 
Where:
and:

Vg
  
Q2  
   
y
 p 

2 vag
 2Q2  
 f0
y
p
Take

2 uag
 2Q1  
 f0
x
p
A

2 vag
 2Q2  
 f0
y
p
B
A B

x y
vag 
  2  2 
 Q1 Q2 
2   uag


    2  2   f 0
 2


y 
y 
p  x
y 
 x
 x
Substitute continuity equation
  2  2 
 Q1 Q2 
 2
    2  2   f 0 2
 2

y 
y 
p
 x
 x
And use vector notation to get:
 2
 Q1 Q2 
2 

    f 0 2   2

p 
y 
 x

COMPARE THIS EQUATION WITH THE TRADITIONAL QG  EQUATION!
1 2
 2 f 02  2 
 
  f 0
   
V g     
2 
 p 
p 

 f0


 
f    2   V g   
p 


We can write the Q-vector form of the QG  equation as:


 2
2 
    f 0 2   2   Q
p 


Where the components of the Q vector are



  Vg
  Vg
 








Q   
    iˆ,  
     ˆj 



 x
 p    y
 p   


  Vg


    ˆ  Vg
    ˆ 

Q 
    i , 
    j 



 x
 p    y
 p   
Using the hydrostatic relationship, we can write Q more simply as:




  Vg
 
R  Vg
Q 
 T iˆ, 
 T  ˆj 
  y
 
p  x
 
 
or in scalar notation as

R  u g T vg T  ˆ  u g T vg T  ˆ 
i  
 j 
Q   


p  x x x y   y x y y  


 2
2 
    f 0 2   2   Q
p 


First note that if the Q vector is convergent



Q  0

 2 Q  0


 2
2 
    f 0 2   0
p 

0
Therefore air is rising when the Q vector is convergent
Q convergence
Q divergence

R  u g T vg T  ˆ  u g T vg T  ˆ 
i  
 j 
Q   


p  x x x y   y x y y  
Let’s go back to our jet entrance region
T
Note that there is no
in this particular jet
y

R  u g T  ˆ  u g T  ˆ 
R T  v g
i , 
 j   
Q   
 
p  x x   y x  
p x  y
 ˆ  u g
i  
  y



R  T  ˆ Vg 
Q 
 k 

p  x  
y 
The Q vectors capture the sense of the
ageostrophic circulation and allow us to
see where the rising motion is occurring
 ˆ
 j 
 
Q convergence
Q divergence
Resolution of the Geostrophic Paradox


R  T   ˆ Vg 
Q 
 k 

p  x  
y 
The Q vectors capture the sense of the
ageostrophic circulation and allow us to
see where the rising motion is occurring
Q vectors diagnose a thermally direct circulation
Adiabatic cooling of rising warm air
Adiabatic warming of sinking cold air
Counteracts the tendency of the geostrophic
temperature advection in confluent flow
Under influence of Coriolis force, horizontal
branches tend to increase shear
Counteracts the tendency of the geostrophic
Momentum advection in the confluent flow
A natural coordinate version of the Q vector
(Sanders and Hoskins 1990)

R  u g T vg T  ˆ  u g T vg T  ˆ 
i  
 j 
Q   


p  x x x y   y x y y  
Consider a zonally oriented confluent
entrance region of a jet where

R  T  vg ˆ vg
Q    
i
p  y  x
y
T
0
x
ˆj 

Use non-divergence of geostrophic wind

R  T  vg ˆ u g
Q    
i
p  y  x
x
or

 R  T   Vg 
 kˆ 
Q  

p  y  
x 
ˆj 

A natural coordinate version of the Q vector
(Sanders and Hoskins 1990)

R  u g T vg T  ˆ  u g T vg T  ˆ 
i  
 j 
Q   


p  x x x y   y x y y  
Consider a meridionally oriented confluent
T
0
entrance region of a jet where
y

R  T  u g ˆ u g
Q    
i
p  x  x
y
ˆj 

Use non-divergence of geostrophic wind

R  T  vg ˆ u g
Q    
i
p  x  y
y
or


R  T   ˆ Vg 
Q 
 k 

p  x  
y 
ˆj 

A natural coordinate version of the Q vector
(Sanders and Hoskins 1990)
Using these two expressions, let’s adopt
A natural coordinate expression for Q


 R  T   Vg 


R  T  ˆ Vg 
ˆ
 k 
Q  
Q 
 k 


p  y  
x 
p  x  
y 
Adopt a coordinate system sˆ, nˆ  where sˆ is directed along the isotherms
nˆ is directed normal to the isotherms


R T  ˆ Vg 
Q
k 

p n 
s 
Q vector oriented perpendicular to the vector change in the geostrophic wind
along the isotherms. Magnitude proportional to temperature gradient and
inversely proportional to pressure.
rising
motion
sinking
motion
Simple Application #1 Train of cyclones and anticyclones

At center of highs and lows: Vg  0

Black arrows: Vg
Gray arrows =
Bold arrows =

Vg
s





V
R T ˆ
g
Q
k



p n 
s 
Note also that because of divergence/convergence, train of cyclones and
anticyclones propagates east along direction of thermal wind
sinking
motion
rising
motion
Simple Application #2 Pure deformation flow with a temperature gradient
Along axis of dilitation

Vg Increases toward east

Black arrows: Vg
Gray arrows =
Bold arrows =

Vg
s



R T ˆ Vg 
Q
k 

p n 
s 
Simple Application #3 Homogeneous warm advection

No variation in Vg along an isotherm

Black arrows: Vg

Vg
0
s

R T
Bold arrows = Q  
p n
Gray arrows =

 Vg 
kˆ 
0

s


No heterogeneity in the warm advection field = No rising motion!
Note that the Q vector form of the QG -equation contains the
deformation terms
(unlike the Sutcliff and Trenberth forms)
And combines the vorticity and thermal advection terms into a single
diagnostic
(unlike the traditional QG -equation)
Sutcliff/Trenberth approximation
Deformation term contribution to 
The along and across-isentrope components of the Q vector
Begin with the hydrostatic equation in potential temperature form 
 
where:    R  p 
fp0  p0 
And the definition of the Q vector:
Substituting:

 f
p
Cv / C p
(which is constant on an isobaric surface)



  Vg
  Vg
 








Q   
    iˆ,  
     ˆj 



 x
 p    y
 p   


 Vg

  Vg
 
ˆ



Q  f  
  i , 
   ˆj 
  y
 
 x
 
 
This expression is equivalent to:

d
Q  f

dtg

d
Q  f

dtg
The Q-vector describes the rate of change of the potential temperature along
The direction of the geostrophic flow
Let’s consider separately the components of Q along and across the isentropes

Qs  alongisentropes

Qn  across isentropes

Qs  alongisentropes

Qn  across isentropes

Qn Is parallel to   and can only affect changes in the magnitude of  

Qs Is perpendicular to   and can only affect changes in the direction of  

  Q    

Qn  
 Qn nˆ

    


  Q  kˆ  
Qs  


 

Q  Qn nˆ  Qs sˆ
 kˆ    Q sˆ
 

s
 

Q  Qn nˆ  Qs sˆ
Returning to QG  equation


 2
2 
    f 0 2   2   Q
p 


Components of vertical motion can be distributed in couplets across (transverse to)
the thermal wind (mean isotherms) and along (shearwise) the thermal wind.
We will see later that the transverse component of Q is related to the dynamics of
frontal zones.