Transcript 08 Ageostrophic wind..
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
d V
f k
ˆ
V
dt
Divide by f and take vertical cross product
f k
ˆ
d V
dt
Note that
k
ˆ
f
f k
ˆ
d V
dt f k
ˆ
k
ˆ
k
ˆ
A
A
V
V
V g
and
f k
ˆ
V g
f k
ˆ
V ag
Black arrows: acceleration vector Gray arrows: ageostrophic wind vector Ageostrophic wind and acceleration vectors in a jetstreak
k f
ˆ
d V 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
k f
ˆ
d V 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
k f
ˆ
d V
dt
V
V g
V ag V
ag
k f
ˆ
V t
V
V
V
p
Let’s only consider the geostrophic contribution to
V
ag
V ag
f k
ˆ
V
g
t
V g
V
g
V
g
p
Let’s only consider the first term
V
ag
k
ˆ
f
V g
t
V
ag
V agT
k
ˆ
f f k
ˆ
V g
t
V g
t
V agT
k
ˆ
f
V g
t
k
ˆ
f
t
Geostrophic wind relationship
k
ˆ
f
f
1 2
p
t
1
f
2
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
Conv pressure increasing under jet right exit region
V g
t V
ag
H jet exit
V g
t V
ag
Div
L
pressure decreasing under jet right exit region
V g
t V
ag
V isal
f k
ˆ
V g
t
f
1 2
p
t
V isal
f
1 2 2
p
t
Convergence of the near surface (ageostrophic) isallobaric wind is related to rising motion
Conv pressure increasing under jet right exit region
V g
t V
ag
H
V g
t
jet exit
V
ag
Div
L
pressure decreasing under jet right exit region
V g
t V
ag
Convergence and divergence of the ageostrophic wind: two simple cases ageostrophic flow in vicinity of jetstreaks and curved flow
k f
ˆ
d V
dt
V
V g
V ag V
ag
k f
ˆ
V t
V
V
V
p
Let’s only consider the geostrophic contribution to
V
ag
V ag
f k
ˆ
V
g
t
V g
V
g
V
g
p
Let’s only consider the second term
V ag
f k
ˆ
V g
V g
V ag
f k
ˆ
V g
V g
This ageostophic wind component is called the inertial-advective wind Let’s expand this:
V IA
k
ˆ
f
u g
u g
x
v g
u g
y
i
ˆ
u g
At black dot:
v g
v x g
0
v g
v g
y
j
ˆ
v g
x
0
V IA
f k
ˆ
u g
u g
x i
ˆ Inertial advective component flows cross jet, consistent with divergence and convergence patterns in jetstreak
Exit region of a jetstreak
V ag
f k
ˆ
V g
V g
This ageostophic wind component is called the inertial-advective wind Let’s expand this:
V IA
k
ˆ
f
u g
u g
x
v g
u g
y
i
ˆ
u g
At black dot:
v g
v x g
0
v g
v g
y
j
ˆ
u g
x
0
V IA
k
ˆ
u g f
v g
x j
ˆ
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 Consider a wind aloft
V
0
V
Consider the vertical shear vector
V
s d V dt
V
0
t
V
0
V s
V
0
d V
s dt
and rewrite:
d V dt
V t
0
V
0
V
0
V s
V
0
d V
s dt
and note that:
d V dt
0
V t
0
V
0
V
0 So we can write: such that
d V
dt
d V
0
dt
d V
s dt
and
d dt
t
V
d V
dt
d V
dt
0
V s
V
0
d V
s dt
d V
dt
d V
dt
0
V s
V
0
d V
s dt
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:
d V
dt
d V
dt
0
V s
V
0
u s
u
0
x
v s
u
0
y
i
ˆ
u s
v
0
x
v s
v
0
y
j
ˆ
d V
dt
d V
dt
0
V s
V
0
u s
u
0
x
v s
u
0
y
i
ˆ
u s
v
0
x
v s
v
0
y
j
ˆ 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
v
0
y
0
d V
dt
d V
dt
0
v s
u
y
0
i
ˆ
d V
dt
d V
dt
0
v s
u
y
0
i
ˆ 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
v s
0
u
0
y
0
v s
u
0
y i
ˆ 0 (black arrow)
k
ˆ
d V
dt
d V
dt
0 Direction of difference in ageostrophic wind between top and bottom of column gray arrow
d V
dt
d V
dt
0
v s v s
0
u
u
0
y y
0
i
ˆ 0
k
ˆ
d V
dt
d V
dt
0
v s
u
0
y i
ˆ 0 (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
d V
dt
d V
dt
0
V s
V
0
d V
s dt
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:
d V
dt
d V
dt
0
d V
s dt
FRONTOGENESIS
d V
dt
d V
dt
0
d V
s dt
Dashed lines: 1000-500 mb thickness Thin gray arrows: Shear vector Black arrow:
d V
dt
d V
dt
0
d V
s dt
Gray arrow:
V ag
ag
0
k
ˆ
d V
dt
d V
dt
0
k
ˆ
d V
s dt
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
d V
dt
d V
dt
0
V s
V
0
d V
s dt
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)
k
ˆ
f
d V
dt
V ag
Use the vector identity: Recall equation for ageostrophic wind
A
B
C
B
A
C
Sutcliff reasoned that:
k
ˆ
d V
dt
k
ˆ
d V
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…..
k
ˆ
d V
dt
k
ˆ
d V
dt
Divergence of ageostrophic wind related to change in vorticity Let’s start with the vorticity equation in 2D (ignoring the tilting terms)
d
dt
d
dt
f
f
V
t f
V
f
p f
f
V
Expand total derivative Or: 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
t
f
V g
g
f
f
0
V
1 2
f
t
V g
g
f
f
0
V
df
0
dt
g
f
1 2
1 2
f
t
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)
f
0
V
V
0
V g
g
where
t
t
t
0
f
V g
0
g
0
f
1 2
f
t
t
0 is the change in thickness between two height surfaces
What is a change in thickness associated with?
d
Let’s find out by expanding total derivative
t
Recall thickness is related To mean temperature between two levels
and
0
d
dt
V
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 2
f
t
1
f
2
u g
x
v g
y
Original equation
f
0
V
V
0
V g
g
f
g
0
f
1 2
f
t
Term on far RHS 1 2
f
t
1
f
2
u g
x
v g
y
Now substitute thermal wind eqn:
f v g
x
f u g
y
f
1 2
t
2
u g v g
v g u g
f
1 2
t
2
u g v g
v g u g
1
f
2
t
x
2 2
y
2 2
u g v g
v g u g
Expand this term, eliminating products of derivatives as small. The terms that are eliminated represent deformation, and they are therefore associated with frontogenesis
g
v g
x
x
f
1 2
t
u g
x
v g
y
v
g
x
u
g
y
u g
y
v g
x
u
g
x
v
g
y
u
g
x
v
g
y
v g
x
u g
y
u
g
y
v
g
x
u g
x
v g
y
Surviving terms in vector form: Terms in yellow represent divergence of mean geostrophic wind and thermal wind Both = 0
f
1 2
t
u g
x
v g
y
v
g
x
u
g
y
u
g
x
v
g
y
v g
x
u g
y
Note that these are expressions of relative vorticity Surviving terms in vector form:
f
1 2
t
V
g
g
V g
g
Substitute:
V g
g
V g g
V g
0
g
0 / / 2 2
V g
V g
V g
0
g
g
g
0 1 2
f
t
V
go
g
V
g
g
0
Original equation
f
0
V
V
0
V g
g
f
V g
0
g
0
f
1 2
f
t
Simplified form of term on RHS 1 2
f
t
V
go
g
V
g
g
0 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
g
1 2
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
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)