ADVENTURE IN SYNOPTIC DYNAMICS HISTORY
How can we tell when and where air is going to go up?
The diagnosis of mid-latitude vertical motions
CHAPTER 6 in Mid-latitude atmospheric dynamics
Why are we interested in vertical motions in the atmosphere?
UNDERSTANDING AGEOSTROPHIC FLOW
The relationship between the ageostrophic wind and the acceleration vector
Vector form of eqn. of motion

dV

  f kˆ  V   
dt
Divide by f and take vertical cross product
kˆ

dV

kˆ
ˆ


  fk  V    
f
dt
f
f
Note that
kˆ

kˆ  kˆ  A   A
kˆ
f


dV
dt

and

kˆ
Vg   
f
 

 V  V g  V ag
Black arrows: acceleration vector
Gray arrows: ageostrophic wind vector
Ageostrophic wind and acceleration vectors in a jetstreak
kˆ
f


dV
dt
 

 V  V g  V ag
Black arrows: acceleration vector
Gray arrows: ageostrophic wind vector
Ageostrophic wind and acceleration vectors in a trough-ridge system
kˆ
f


dV
dt
 

 V  V g  V ag
Convergence and divergence of the ageostrophic wind: two simple cases
ageostrophic flow in vicinity of jetstreaks and curved flow

ˆ
 

k dV

 V  V g  V ag
f
dt


ˆ



k  V
V 


V ag 

 V  V  

f  t
 p 

Let’s only consider the geostrophic contribution to V ag

V ag




V g 
kˆ   V g



 Vg  Vg  
f   t
 p 
Let’s only consider the first term

V g

kˆ
V ag 

f
t

V g

Geostrophic
kˆ
V ag 

wind relationship
f
t



kˆ  V g
kˆ   kˆ
1

     
V agT 




2

f
t
f  t  f
f
t


ˆ

k V g
1
p
V agT 



2
f
t
f
t
Pressure coordinates
Height coordinates
This component of the ageostrophic wind is called the isallobaric wind
because the ageostrophic wind flows in the direction of the gradient in the
pressure tendency
pressure
increasing under
jet right exit region

V ag
Conv
jet
exit
Div
pressure
decreasing under
jet right exit region

V g

V g

V g
t
t
t
H

V ag
L

V ag
V isal 
kˆ
f


V g
t

1
f
2

p
t

1
2 p
  V isal  

2
f
t
Convergence of the near surface (ageostrophic) isallobaric wind is related to rising motion
Conv
pressure
increasing under
jet right exit region

V ag
jet
exit
Div
pressure
decreasing under
jet right exit region

V g

V g

V g
t
t
t
H

V ag
L

V ag
Convergence and divergence of the ageostrophic wind: two simple cases
ageostrophic flow in vicinity of jetstreaks and curved flow

ˆ
 

k dV

 V  V g  V ag
f
dt


ˆ



k  V
V 


V ag 

 V  V  

f  t
 p 

Let’s only consider the geostrophic contribution to V ag

V ag




V g 
kˆ   V g



 Vg  Vg  
f   t
 p 
Let’s only consider the second term


kˆ 
V ag 
Vg  Vg
f


kˆ 
V ag 
Vg  Vg
f
This ageostophic wind component is called the inertial-advective wind
Let’s expand this:

u g
u g 
v g
v g  

kˆ  
ˆ
i   u g
 ˆj 
V IA 
   u g
 vg
 vg



f 
x
y 

x

y

 
At black dot: v g  0
v g
x
0

u g
kˆ
V IA 
 ug
iˆ
f
x
Inertial advective component flows
cross jet, consistent with divergence
and convergence patterns in jetstreak
Exit region of a jetstreak


kˆ 
V ag 
Vg  Vg
f
This ageostophic wind component is called the inertial-advective wind
Let’s expand this:

u g
u g 
v g
v g  

kˆ  
ˆ
i   u g
 ˆj 
V IA 
   u g
 vg
 vg



f 
x
y 

x

y

 
At black dot: v g  0
u g
x
0

v g
kˆ
ˆj
V IA 
 ug
f
x
Inertial advective component flows
in direction of geostrophic wind,
consistent with supergeostrophic flow
in crest of ridge
Exit region of a jetstreak
Sutcliff’s (1939) expression for ageostrophic divergence

Consider a surface wind V 0

V
Consider a wind aloft

Consider the vertical shear vector V s



such that d V  d V 0  d V s
dt
dt
dt
Expand expression in orange box:



V0
dVs

 V0  V s   V0 
dt
t
dt

dV


and
d
dt


t
V 
and rewrite:



V0
dVs

 V0   V0  V s   V0 
dt
t
dt

dV
and note that:


 dV 
V0

 
 V0   V0
 dt 

t

0
So we can write:



 dV 
dVs
  V s   V0 

 dt 
dt
dt

0

dV



 dV 
dVs
  V s   V0 

 dt 
dt
dt

0

dV
The difference between the acceleration of the wind aloft and the acceleration of the wind
at the surface is related to the shear over the surface wind gradient and the rate of change
of the wind shear following parcel motion. (are you rather confused??)
Let’s take it apart and try to understand a simple example
Examine first term on RHS:

dV


 dV 
 u 0
u 0  ˆ  v0
v0  ˆ
  V s   V0   u s


 j


v
i

u

v
s
s
s





dt
x
y 
x
y 


 dt  0

dV


 dV 
 u 0
u 0  ˆ  v0
v0  ˆ
  V s   V0   u s


 j


v
i

u

v
s
s
s





dt
x
y 
x
y 


 dt  0
Dashed lines: 1000-500 mb thickness
(mean temperature in 1000-500 mb layer)
Solid lines: Isobars
Little arrows: Geostrophic wind

Black arrow: V s   V 0
Gray arrow: Difference between upper and lower
level ageostrophic wind
Red arrow: Shear vector
Shear northward along direction of mean isentropes
At center of low: u s  0

dV
v0
y

 dV 
u 0 ˆ
  vs

i


dt
y
 dt  0
0

dV

 dV 
u 0 ˆ
  vs

i


dt
y
 dt  0
Dashed lines: 1000-500 mb thickness
(mean temperature in 1000-500 mb layer)
Solid lines: Isobars
Little arrows: Geostrophic
wind

Black arrow: V s   V 0
Gray arrow: Difference between upper and lower
level ageostrophic wind
Red arrow: Shear vector
vs  0
u 0
y
0


 dV  dV  
 
kˆ  



 dt
 dt  0 
vs
u 0 ˆ
i 0
y
(black arrow)
Direction of difference in
ageostrophic wind between
top and bottom of column
gray arrow

dV

 dV 
u 0 ˆ
  vs

i


dt
y
 dt  0
u 0
0
vs  0
y


 dV  dV  
 
kˆ  



 dt
 dt  0 
vs
u 0 ˆ
i 0
y
(black arrow)
Direction of difference in
ageostrophic wind between
top and bottom of column
gray arrow
Ageostrophic wind at surface at low center = 0
Ageostrophic wind aloft points south
Aloft: wind diverges at D, convergences at C
Low propagates toward D, or along the direction
of the geostrophic shear (mean isotherms)
THE SEA-LEVEL PRESSURE PERTURBATION PROPAGATES IN
THE DIRECTION OF THE THERMAL WIND VECTOR



 dV 
dVs
  V s   V0 

 dt 
dt
dt

0

dV
The difference between the acceleration of the wind aloft and the acceleration of the wind
at the surface is related to the shear over the surface wind gradient and the rate of change
of the wind shear following parcel motion. (are you rather confused??)
Let’s take it apart and try to understand a second simple example
Examine second term on RHS:


 dV 
dVs
 

 dt 
dt
dt

0

dV


 dV 
dVs
 

 dt 
dt
dt

0

dV
FRONTOGENESIS
Dashed lines: 1000-500 mb thickness
Thin gray arrows: Shear vector
Black arrow:

dV

 dV

 dt
dt



dVs
 

dt
0
Gray arrow:


V ag  V ag
 
0



 dV  dV  
d
V
s
   kˆ 
 kˆ  



 dt

dt
 dt  0 

Gray arrow is the difference in the ageostrophic flow between upper and lower troposphere
Air diverges aloft on warm side of front: rising motion on warm side
Air converges aloft on cold side of front: sinking motion on cold side
1939: First dynamical understanding of the effect of
frontogenesis on vertical circulations about fronts



 dV 
dVs
  V s   V0 

 dt 
dt
dt

0

dV
Consider the historical significance of this equation:
In 1939, when Sutcliff published this result, the U.S. Military
weather forecasters were just beginning to launch
rawinsondes around the country. There were no computers
or forecast models.
This relationship allowed forecasters, from measurements of
temperature at two levels and the sea level pressure field, to
forecast the direction of movement of highs and lows!
The relationship also allowed forecasters to diagnose where
upward motion would occur by comparing the 1000-500 mb
thickness patterns at two times.
The Sutcliffe Development Theorem (1949)
kˆ

f

dV
dt

 V ag
Use the vector identity:
Recall equation for ageostrophic wind
  
 
A  B  C  B  A C
and apply  operator:
Sutcliff reasoned that:
   kˆ 

dV
dt
 kˆ   

dV
dt
On an f plane (f constant) the divergence
Of the ageostrophic wind is related to
Changes in the vertical component of vorticity
…and sought to understand how vorticity may be used as a diagnostic
tool to determine where divergence, and hence rising motions might occur
Let’s look at Sutcliff’s reasoning…..
   kˆ 

dV
 kˆ   
dt

dV
Divergence of ageostrophic wind related to change in vorticity
dt
Let’s start with the vorticity equation in 2D (ignoring the tilting terms)
d

d   f
dt
   f
t

dt


    f   V

   f
 V     f   
p


    f   V
Expand total derivative
Now assume 1) vorticity and horizontal winds are geostrophic
2) vertical advection of vorticity is negligible
3) relative vorticity can be neglected in divergence term
  g  f
t
Or:
1
f

2



 V g    g  f    f 0   V
df
0
dt

t


 V g    g  f    f 0   V
g 
1
f
 
2
1

2

t
f


 V g    g  f    f 0   V
Sutcliff’s idea: Consider difference in divergence between the top and
bottom of an air column (say at 300 and 700 mb)




1 2  
f 0   V    V 0  V g    g  f   V g 0    g 0  f   
f
t


 
where
t


t

0
t


t
   0 
is the change in thickness
between two height surfaces
What is a change in thickness associated with?
Let’s find out by expanding total derivative
d 
dt
Recall thickness is related
To mean temperature between
two levels  and  0
 
d 

 

 V     
t
dt
p
Diabatic heating or cooling
Thickness advection
Vertical advection
(adiabatic heating or cooling)
Sutcliff 1) ignored diabatic cooling as small,
2) ignored vertical advection to simplify the problem
3) assumed V = Vg = mean geostrophic wind in layer

1
f

2
 
t

 
  
2

  u g
 vg
f
x
y 

1
Original equation




1 2  
f 0   V    V 0  V g    g  f   V  g 0  f   
f
t

Term on far RHS


1

2
f
 
t

f v g 
Now substitute thermal wind eqn:

1
f
 
  
2

  u g
 vg
f
x
y 

1

2
 
t
 
x
 f u g 
 
y
  u g v g  v g u g 
2
1


2
f

1
f

2
 
t
  u g v g  v g u g 
 
2
2
 2

  2 
2
t
y
 x

 u g v g  v g u g 


Expand this term, eliminating products of derivatives as small. The terms that are
eliminated  u  v  represent deformation, and they are therefore associated with frontogenesis
g
x

1
f

g
x
2


    v g  u g
 
  u g
 vg

t

x

y

x
y


 


   v g  u g


 u g
 
 vg


x

y

x
y


 

    u g  v g
   u g
 
 vg



y

x

x
y

 
 

   u g  v g

   u g
 
 vg



y

x

x
y

 
Surviving terms in vector form:








Terms in yellow
represent divergence
of mean geostrophic
wind and thermal wind
Both = 0

1

f
2


    v g  u g
 
  u g
 vg

t
x
 y   x
y

 
 

   v g  u g
g
   u g
 

v


x
 y   x
y
 




Note that these are expressions of relative vorticity
Surviving terms in vector form:

1

f
2
 
t
Substitute:




 V g    g  V g    g




Vg  Vg  Vg0 / 2


g
 

   
  
  V go   
f
 t 
1

g
2
g

g0




V g  V g  V g 0
/ 2
 g   g   g 0

 Vg  
g0
Original equation




1 2  
f 0   V    V 0  V g    g  f   V g 0    g 0  f   
f
t


Simplified form of term on RHS

   
  
  V go   
f
 t 
1
2
g

 Vg  
g0
Plug it in:






f 0   V    V 0  V g    g  f   V g 0    g 0  f   V g 0    g  V g    g 0


Reduce right hand side




f 0   V    V 0   V g  V go    g 0   g  f

And finally….







f 0   V    V 0  V     g 0   g  f






f 0   V    V 0  V     g 0   g  f



Synoptic scale vertical motions:
(the result of greater divergence or convergence aloft in an air column)
can be diagnosed on weather maps
How? Plot geopotential height field at two levels

Graphically subtract them to get thermal wind vector V 
Use same fields to determine vorticity at each level (using  g 
1
f
  )
add them up and determine advection of total vorticity by thermal wind
Today this all seems like too much work!!!
But in 1949, the technique revolutionized synoptic meteorology
2
300-700 mb thickness
Vorticity term
in Sutcliff equation
Sutcliff vertical motion
at 500 mb (microbars/s)
actual vertical motion
At 500 mb (microbars/s)