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