adventure in synoptic dynamics history. how can we tell when and where air is going to go up?
TRANSCRIPT
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
Why are we interested in vertical motions in the atmosphere?
CHAPTER 6 in Mid-latitude atmospheric dynamics
Vkfdt
Vd
ˆ
The relationship between the ageostrophic wind and the acceleration vector
f
kVkf
f
k
dt
Vd
f
k ˆˆ
ˆˆ
agg VVVdt
Vd
f
k
ˆ
Vector form of eqn. of motion
Divide by f and take vertical cross product
f
kVg
ˆAAkk ˆˆNote that and
UNDERSTANDING AGEOSTROPHIC FLOW
Black arrows: acceleration vectorGray arrows: ageostrophic wind vector
Ageostrophic wind and acceleration vectors in a jetstreak
agg VVVdt
Vd
f
k
ˆ
Black arrows: acceleration vectorGray arrows: ageostrophic wind vector
Ageostrophic wind and acceleration vectors in a trough-ridge system
agg VVVdt
Vd
f
k
ˆ
Convergence and divergence of the ageostrophic wind: two simple cases
ageostrophic flow in vicinity of jetstreaks and curved flow
agg VVVdt
Vd
f
k
ˆ
p
VVV
t
V
f
kVag
ˆ
p
VVV
t
V
f
kV g
ggg
ag
ˆ
Let’s only consider the geostrophic contribution to agV
Let’s only consider the first term
t
V
f
kV gag
ˆ
tff
k
tf
k
t
V
f
kV gagT
2
1ˆˆˆ
t
p
ft
V
f
kV gagT
2
1ˆ
t
V
f
kV gag
ˆ Geostrophic
wind relationship
Pressure coordinates
Height coordinates
This component of the ageostrophic wind is called the isallobaric windbecause the ageostrophic wind flows in the direction of the gradient in thepressure tendency
H LagV
jetexit
DivConv
t
Vg
pressureincreasing under
jet right exit region
pressuredecreasing under
jet right exit region
t
Vg
agV
t
Vg
agV
t
p
ft
V
f
kV gisal
2
1ˆ
t
p
fVisal
2
2
1
Convergence of the near surface (ageostrophic) isallobaric wind is related to rising motion
H LagV
jetexit
DivConv
t
Vg
pressureincreasing under
jet right exit region
pressuredecreasing under
jet right exit region
t
Vg
agV
t
Vg
agV
p
VVV
t
V
f
kV g
ggg
ag
ˆ
Let’s only consider the geostrophic contribution to agV
Let’s only consider the second term
ggag VVf
kV
ˆ
Convergence and divergence of the ageostrophic wind: two simple cases
ageostrophic flow in vicinity of jetstreaks and curved flow
agg VVVdt
Vd
f
k
ˆ
p
VVV
t
V
f
kVag
ˆ
ggag VVf
kV
ˆ
Let’s expand this:
jy
vv
x
vui
y
uv
x
uu
f
kV g
gg
gg
gg
gIAˆˆ
ˆ
This ageostophic wind component is called the inertial-advective wind
Exit region of a jetstreak
At black dot: 0gv 0x
vg
ix
uu
f
kV g
gIAˆ
ˆ
Inertial advective component flowscross jet, consistent with divergenceand convergence patterns in jetstreak
ggag VVf
kV
ˆ
Let’s expand this:
jy
vv
x
vui
y
uv
x
uu
f
kV g
gg
gg
gg
gIAˆˆ
ˆ
This ageostophic wind component is called the inertial-advective wind
Exit region of a jetstreak
At black dot: 0gv 0x
ug
jx
vu
f
kV g
gIAˆ
ˆ
Inertial advective component flowsin direction of geostrophic wind,
consistent with supergeostrophic flowin crest of ridge
Sutcliff’s (1939) expression for ageostrophic divergence
Consider a surface wind
dt
Vd
dt
Vd
dt
Vd s
0
0V
Consider a wind aloft V
Consider the vertical shear vector sV
such that
Vtdt
dand
Expand expression in orange box:
dt
VdVVV
t
V
dt
Vd ss
000
dt
VdVVVV
t
V
dt
Vd ss
0000
and rewrite:
and note that:
000
0
VVt
V
dt
Vd
So we can write:dt
VdVV
dt
Vd
dt
Vd ss
0
0
dt
VdVV
dt
Vd
dt
Vd ss
0
0
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 changeof 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:
jy
vv
x
vui
y
uv
x
uuVV
dt
Vd
dt
Vdsssss
ˆˆ 00000
0
jy
vv
x
vui
y
uv
x
uuVV
dt
Vd
dt
Vdsssss
ˆˆ 00000
0
Dashed lines: 1000-500 mb thickness (mean temperature in 1000-500 mb layer)Solid lines: IsobarsLittle arrows: Geostrophic windBlack arrow:Gray arrow: Difference between upper and lower level ageostrophic windRed arrow: Shear vector
0VVs
Shear northward along direction of mean isentropes
At center of low: 0su 00 y
v
iy
uv
dt
Vd
dt
Vds
ˆ0
0
Dashed lines: 1000-500 mb thickness (mean temperature in 1000-500 mb layer)Solid lines: IsobarsLittle arrows: Geostrophic windBlack arrow:Gray arrow: Difference between upper and lower level ageostrophic windRed arrow: Shear vector
0VVs
iy
uv
dt
Vd
dt
Vds
ˆ0
0
0sv 00 y
u0ˆ0
iy
uvs (black arrow)
0
ˆdt
Vd
dt
Vdk
Direction of difference inageostrophic wind betweentop and bottom of column gray arrow
iy
uv
dt
Vd
dt
Vds
ˆ0
0
0sv 00 y
u0ˆ0
iy
uvs (black arrow)
0
ˆdt
Vd
dt
Vdk
Direction of difference inageostrophic wind betweentop 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 directionof the geostrophic shear (mean isotherms)
THE SEA-LEVEL PRESSURE PERTURBATION PROPAGATES IN THE DIRECTION OF THE THERMAL WIND VECTOR
dt
VdVV
dt
Vd
dt
Vd ss
0
0
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 changeof 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:
dt
Vd
dt
Vd
dt
Vd s
0
dt
Vd
dt
Vd
dt
Vd s
0
Dashed lines: 1000-500 mb thicknessThin gray arrows: Shear vector
Black arrow:
Gray arrow:
dt
Vd
dt
Vd
dt
Vd s
0
dt
Vdk
dt
Vd
dt
VdkVV s
agag
ˆˆ
00
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
FRONTOGENESIS
1939: First dynamical understanding of the effect of frontogenesis on vertical circulations about fronts
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 whereupward motion would occur by comparing the 1000-500 mb thickness patterns at two times.
dt
VdVV
dt
Vd
dt
Vd ss
0
0
The Sutcliffe Development Theorem (1949)
agVdt
Vd
f
k
ˆ
Recall equation for ageostrophic wind
Use the vector identity: CABCBA
dt
Vdk
dt
Vdk
ˆˆ
and apply operator:
On an f plane (f constant) the divergenceOf the ageostrophic wind is related toChanges in the vertical component of vorticity
Sutcliff reasoned that:
…and sought to understand how vorticity may be used as a diagnostictool to determine where divergence, and hence rising motions might occur
Let’s look at Sutcliff’s reasoning…..
dt
Vdk
dt
Vdk
ˆˆ
Let’s start with the vorticity equation in 2D (ignoring the tilting terms)
Divergence of ageostrophic wind related to change in vorticity
Vfdt
fd
dt
d
Vfp
ffV
t
f
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
VffVt
fgg
g
0
Or: VffVtf gg
021
0dt
df
21fg
Sutcliff’s idea: Consider difference in divergence between the top and bottom of an air column (say at 300 and 700 mb)
VffVtf gg
021
tf
fVfVVVf gggg
2
0000
1
00
tttt
whereis the change in thicknessbetween two height surfaces
What is a change in thickness associated with?
Let’s find out by expanding total derivative
pV
dt
d
t
dt
d
Recall thickness is relatedTo mean temperature betweentwo levels 0 and
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
yv
xu
ftf gg
22 11
tf
fVfVVVf ggg
2
000
1Original equation
yv
xu
ftf gg
22 11Term on far RHS
Now substitute thermal wind eqn: xvf g
yuf g
gggg uvvutf
221
Expand this term, eliminating products of derivatives as small. The terms that areeliminated represent deformation, and they are therefore associated with frontogenesis
y
v
x
u
xv
yu
y
u
x
v
yv
xu
y
v
x
u
xv
yu
y
u
x
v
yv
xu
tf
gggg
gggg
gggg
gggg
21
Terms in yellowrepresent divergenceof mean geostrophicwind and thermal windBoth = 0Surviving terms in vector form:
x
v
x
u gg
gggg uvvutf
221
gggg uvvuyxtf
2
2
2
221
y
u
x
v
yv
xu
y
u
x
v
yv
xu
tfgg
gggg
gg
21
Surviving terms in vector form:
Note that these are expressions of relative vorticity
gggg VVtf
21
021
ggggo VVtf
2/0ggg VVV
0ggg VVV
2/0ggg 0ggg Substitute:
tf
fVfVVVf gggg
2
0000
1Original equation
Simplified form of term on RHS
Plug it in:
000000 gggggggg VVfVfVVVf
Reduce right hand side
fVVVVf gggog 000
And finally…. fVVVf gg 000
021
ggggo VVtf
fVVVf gg 000
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
Use same fields to determine vorticity at each level (using )
V
21fg
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
actual vertical motionAt 500 mb (microbars/s)
Sutcliff vertical motionat 500 mb (microbars/s)300-700 mb thickness
Vorticity termin Sutcliff equation