Uccellini and Johnson (1979)

Uccellini, L. W., and D. R. Johnson, 1979: The Coupling of Upper and Lower Tropospheric Jet Streaks and Implications for the Development of Severe Convective Storms. Mon. Wea. Rev. 107, 682–703.

Purpose of paper:

To determine how the transverse ageostrophic circulation about an upper tropospheric jetstreak influences the intensity of the low level jet and the development of severe convection

Typical severe weather scenario with upper and lower level jets

Methods:

1) Theory2) Numerical model3) Observation

indirect circulation associated with the exit region of a jetstreak

Contours (vertical motion in b/s)Arrows (ageostrophic circulation)Number under arrows (ageostrophic wind (m/s))

OUR GOAL:

Derive an equation that relates the ageostropic wind

on a low level theta surface to the pressure tendency. Low level pressure changes are forced by the upper level

jetstreak.

The equation we seek will theoretically relate an

accelerating low level jet to the ageostrophic circulation

induced by the upper level jetPs

decreasingPs

increasing

JET

THEORY

The theoretical development in this paper is cast in isentropic coordinates

gg

gg U

dt

dUU

t

U

dt

Ud

dt

Ud

U

dt

dUU

t

U

dt

Ud

dt

Ud

The (vector) expansion of the total derivative in isentropic coordinates

The geostrophic momentum approximation: We approximate U by Ug except in the advection term

We will apply the geostrophic momentum approximation in this paper: Assume that the wind can be approximated by its geostrophic value except when considering advection.

gg

gg U

dt

dUU

t

U

dt

Ud

dt

Ud

From before:

Use vector form of momentum equation

fvxdt

du

gfvx

)( gvvfdt

du

Recall in scalar notation

dt

UdkfUUU gag

1

agvdt

duf 1

Substitute from (1) into (2)

(1)

(2)

ggg

gag

U

dt

dkUkUUkU

t

UkfU

1

dt

Ud

Assume adiabatic flow and neglect second term

g

gag UkU

t

UkfU

1

0dt

d smallk

g

gag UkU

t

UkfU

1

Recall from ATMOS 403 that the geostrophic wind on a theta surface is related to the gradient of the Montgomery stream function: kfU g

1

With the assumptions made above, the ageostrophic wind is related to

1. the rate of change of the pressure gradient force (i.e. geostrophic wind) on a theta surface (the isallobaric component)

2. Advection/inertial forces

where the Montgomery stream function is given by

Put in the isallobaric term and it becomes….. gU

gzTC p

t

fU iag2

,

Isallobaric contribution to the ageostrophic wind

Hydrostatic equation in isentropic coordinates

pcR

p p

pc

/

0

L

S

p

dp

pc

cR

pSL

/

0

Integration of hydrostatic equation from surface to theta surface (L)

tp

pcd

t

p

p

pc

pc

Rc

tt

p

L

S

p cR

sp

cR

pp

pSL

/

0

1/

00

Time derivative of equation above:

tp

pcd

t

p

p

pc

pc

Rc

tt

p

L

S

p cR

sp

cR

pp

pSL

/

0

1/

00

Find expression for this term

ssps gzTc

scR

ssp

s gztp

p

tc

t

p

/

0

Substitute equation for theta

pcR

sss p

pT

/

0

And take derivative

01

00

/

0

t

p

p

p

pc

Rc

tp

pc

ts

c

R

s

p

sps

cR

sp

sp

p

Substitute in top equation: last RHS term in top cancels first RHS term on bottom

From last page…

t

p

p

p

p

Rd

t

p

p

pc

p

R

ts

c

R

ss

cR

pL

pL

S

p

1

00

1/

00

Simply by using expression for s

t

p

p

RTd

t

p

pp

pcR

tss

cR

pL L

S

p

0

/

0

1

t

fU iag2

,

Isallobaric contribution to the ageostrophic windFrom a while ago

Combine equations:

L

S

p

dt

p

pp

pc

t

p

p

RT

f

RU

cR

pss

iag

1/

002,

L

S

p

dt

p

pp

pc

t

p

p

T

f

RU

cR

pss

iag

1/

002,

The isallobaric component of the

ageostrophic circulation on an

isentropic surface L…

..is related tothe gradient

of the surfacepressure

tendency ..

..and the integrated pressure tendencyBetween the surface and L

s,1s,2

L

0

t

ps

t

p

1

0

t

ps

JETEXIT

t

p

2

iagU ,

An existing low level jet will accelerate in this region, increasing shear and potential for severe weather

Model

Hybrid isentropic/sigma coordinate model used to simulate jetstreak circulations

L

lower 200 mb = coordinates

rest of atmosphere = coordinates

Initial conditions: functions designed to produce jetstreak

Mass flux divergence ( 10 g m-2 s-1)

in each quadrant of jetstreak as a function of altitude () from model

Solid (total), ageostrophic (dashed), geostrophic (dot dashed)

Air exiting column

Air entering column

ageostrophic component dominates

g

gag UkU

t

UkfU

1

Isallobaric term Inertial/advective term

Streamline analysis at the level of the jet streak due to the inertial advective term

At the level of the jetstreak, the pressure tendency is small (weak isallobaric effect), but advection/coriolis force effects are large

L

S

p

dt

p

pp

pc

t

p

p

RT

f

RU

cR

pss

iag

1/

002,

At low levels, the isallobaric effect dominates, creating the ageostrophic acceleration of the winds toward the north under the exit region of the jet

Ageostrophic flow forced by along stream variations in PGF at jetstream level

Isallobaric ageostrophic wind on a lower isentropic surface beneath jet

Entrance Exit

P P

Surface pressure falls in exit region lead to ascent of theta surfaces and isallobaric acceleration upward and northward along potential temp surface

Differential diabatic heating has the same effect

Cool here

Heat here(latent heatin convection)

Heating transportsmass from belowto above L

P on isentropic surf

P on isentropic surf

An example case study:

This study chosen because there was a jetstreak in the upper troposphere, but no low level jet in the lower atmosphere.

The development of the LLJ could be attributed to the isallobaric accelerations associated with the jetstreak circulation.

Surface charts for the 12 hour period

Radar depictions: Cloud tops in hundreds of feet RW = rainshowers

TRW=thunderstorms

330 K isotachs and Mongomery streamfunction

300 K isotachs and Mongomery streamfunction

300 K pressure and mixing ratio

Jetstreak propagating eastward

No cross-jet flow here

SW 20 m/s LLJ between Kentucky and Pennsylvania flowing from 900 to 700 mb

Moist air intrusion

Downward extension ofUpper jetstreak

Another view of upper jetstreak and LLJ on isobaric surfaces

Note angle between the isotachs and height contours:

LLJ represents ageostrophic flow associated with isallobaric forcing

340 K

300 K

Surface

340 K

300 K

Surface

Mass tendencies in layers bounded by isentropes or surface

Green: increasing with timeRed: decreasing with time

12Z 10 May

00Z 11 May

Note the four quadrant jetstreak pattern

Actual data in cross section illustrating same effect

305 K

325 KPotential

temperature

Wind speed

J

J

L

S

p

dt

p

pp

pc

t

p

p

T

f

RU

cR

pss

iag

1/

002,

Evaluation of the integrated pressure tendency term in the equation for the isallobaric ageostrophic wind

on 300 K surface

iagU ,

L

S

p

dt

p

pp

pc

t

p

p

T

f

RU

cR

pss

iag

1/

002,

Term 1: Isallobaric wind

Term 3: Integrated

pressure tendencyTerm 2: Surface

pressure tendency

Further illustrates lower level ageostrophic flow induced by upper level jet streak

Cross sections from Omaha, Nebraska to Tetersboro, New Jersey

Normal wind

Moisture flux

Sensible Heat flux

Implications of coupled jet streaks for convective

storm development

Intensification of the LLJ by isallobaric forcing:

1) significantly increases lower atmospheric moisture flux

2) Significantly increases lower atmospheric sensible heat flux

3) enhances low level shear

THEREFORE:

enhances environment for severe storm development