Keyser and Shapiro (1986)

Keyser, D. and M. A. Shapiro, 1986: A review of the structure and dynamics of upper-level frontal zones. Mon. Wea. Rev. 114, 452–499.

s

nTime = t

T

T-

T- 2TT- 3T

T- 4T

T- 5T

T- 6T

T- 7T

T- 8T

s

nTime = t + t

TT- T- 2TT- 3TT- 4TT- 5TT- 6TT- 7TT- 8T

pnnn

v

ndt

d n Fronts exist in a kinematic sense due to deformation in the presence of a thermal gradient and tilting of vertical thermal gradients

Before

n

z

After

n

z

nn

vn

pn

p

VkVff

dt

dpp

Absolute vorticity is generated along parcel trajectories by horizontal convergence and tilting of vertical shear.

CONVERGENCE

Early models of the structure of upper level fronts1937 1949

1952 1959

Bjerknes and Palmen (1937)Palmen and Nagler (1949)Berggren (1952)Reed and Danielson (1959)

Fig. 1

Note different conceptual ideas of the interface of the front with the stratosphere

Temperature

Pot. Temp.

Note folded tropopause

Front envisioned to separate polar and tropical air at all altitudes – stratosphere distinct from front, and isolated from troposphere except for diffusion

-- Vertical extension of surface front Fig. 2Bjerknes and Palmen (1937)

Geostrophic wind normal to cross section, temperature

Cross section normal to front on 5 Feb 1947

Folded tropopause is replaced by a “break region” separating the tropospheric frontal layer and the tropopause over the tropical and polar air

Palmen and Nagler (1949)

Front is a zone of concentrated cyclonic and vertical shear

Jet over frontal zone

Fig. 3

Berggren (1952)

Potential temperature and observed wind speed

Cross section normal to front on 9 Nov 1949

Tropospheric frontal zone is extended into the stratosphere

Frontal zone defined by strong cyclonic wind shear

Fig. 4

Reed and Danielson (1959)Geostrophic wind normal to cross section Temperature

Potential VorticityPotential Temperature

Tropopause on polar side joined to base of frontal zone

Tropopause on tropical side joined to top of frontal zone Frontal zone region of:

• High potential vorticity (stratospheric intrusion of air)• Strong cyclonic and vertical shear• Sharp temperature and potential temperature gradient• Isentropes approximately parallel to frontal surface

p

fP

Fig. 5

Stratospheric air

Difference in thinking:

NEW: Upper tropospheric fronts:

Separate stratospheric and tropospheric air Form as a result of tilting of the horizontal temperature gradient and vorticity

OLD: Upper tropospheric fronts

Separate tropical and polar air Form as a result of confluence between polar and tropical airmasses

p

VkVff

dt

dpp

pnnn

v

ndt

d n

In this view: Upper and lower tropospheric fronts can arise independently Upper tropospheric fronts do not have to extend to the surface

Absolute Geostrophic Momentum

fyum g ug = along front (x) component of geostrophic windf = Coriolis parametery = cross front coordinate, positive toward colder air

Relationship of to 3-D absolute geostrophic vorticity vector

into screen (along front) m1 m2

Frontal zone

p

mk

y

mjm

2

i

2g2gm

y

mk

p

mjmig

22

If there are no along front variations and the front is straight

Vertical component of vorticity

Horizontal component of vorticity

Recent advances in our understanding of upper level fronts

1

2

3

4

5

y

m1 m2 m3 m4 m5

x

p

Barotropic Atmosphere (no temperature gradient)

m surfaces and surfaces in a barotropic and baroclinic environment

1

2

3

4

5

y

m1 m2 m3 m4 m5

x

p

Baroclinic Atmosphere (temperature gradient)

Because of temperature gradientgeostrophic wind increases with heightAnd m surfaces tilt since m = ug - fy

m only a function of f along y direction

Relationship of to potential temperaturem

yp

m

p

v

c

c

p

p

fp

R

0

0

where

Pot. Temp and wind speed Potential Vorticity

Absolute Angular Momentum Absolute Angular Momentum/Pot Temp

TropopauseAnd fronts

Cyclonic Shear Boundary

Extremely high resolution measurements of frontal structure made with a research aircraft supplemented by sondes (Shapiro 1981)

In pressure coordinates, potential vorticity takes the following form:

pp yp

m

py

mP

2

Potential vorticity is:

Large and positive (stable stratospheric air) where the area of the boxes formed by the intersection of m and lines is small

Small and positive (stable tropospheric air) where the area of the boxes is large

Negative (either inertially, convectively, and/or symmetrically unstable) when the slope of the lines exceeds the slope of the m lines

For adiabatic processes (no friction, no diabatic heating, no mixing) potential vorticity is conserved.

Question: Where does the high values of potential vorticity at the inflection point come from?

Clear air turbulence (CAT):

The shear zones associated with fronts are zones of extreme CAT

CAT mixes warm air downward above the level of maximum winds and cold air upward below the level of maximum winds

CAT limits frontal scale collapse to about 100 km

Warming rates due to vertical flux of heat

Upper level frontal zones, potential vorticity anomalies and ozone concentrations

Potential temperature and winds Potential vorticity Ozone

Ozone is a tracer on the time scale of frontogenesis.

Ozone from the stratosphere can intrude all the way to the ground within frontal zones in exceptionally strong fronts!

CAT and Tropospheric-Stratospheric exchange: Chloroflorocarbons go up and radioactivity from the 1950s nuclear bomb tests comes down!

Consider a jetstreak propagating around and through the base of a trough:

Baroclinic instability-wave amplifying

Warm advection

Cold Advection

Baroclinic wave maximum intensity

Cold Advection

Baroclinic wave weakening

What is the impact of cold and warm advection on the ageostrophic circulation about the front and upper level frontogenesis?

We will first look at numerical simulations for

a) a straight jet with confluence only b) a straight jet dominated by warm advection aloftc) a straight jet dominated by cold advection aloft

Then look at theory for these situations and expected patterns of vertical motion

Then look at observations to see how these predictions hold up

Then add in the effect of flow curvature

Procedure:

(Note: Paper does this in the opposite way (observation, theory, simulation), but it is more difficult to understand

Y (km) Y (km)

Y axis Y axis

Y axis Y axis

Pure confluenceacting on gradient

Shear acting on gradientWinds stronger on right leadingTo dominance of cold advection

Shear acting on gradientWinds stronger on left leadingTo dominance of warm advection

Model is 2-D, but the assumed flow that is the basis for the 2D simulation is shown here for the 438 mb level

Symmetric direct circulation about the front with warm air rising and cold air sinking.

Westerly jet on warm side, easterly at low levels on cold side. Air parcels converge and descend in frontal zone

and cross front geostrophic wind

0 hr

48 hr

and along front geostrophic wind

and ageostrophic circulation

24 hr

48 hr

and 48 hr trajectories

Direct circulation displaced toward cold air so that warm air rising along part of frontal zone.

Westerly jet on warm side, easterly at low levels on cold side. Shear zone more defined. Air parcels descend in frontal zone on cold side, ascend on warm side.

and cross front geostrophic wind

0 hr

48 hr

and along front geostrophic wind

and ageostrophic circulation

24 hr

48 hr

and 48 hr trajectories

Direct circulation displaced toward warm air so that warm air sinks into frontal zone

This leads to a frontal split, with the upper level front distinct from the lower level front.

and cross front geostrophic wind

0 hr

48 hr

and along front geostrophic wind

and ageostrophic circulation

24 hr

48 hr

and 48 hr trajectories

Note that the model solutions we just examined are all for confluent flow. Therefore, they apply to a jetstreak’s entrance region. The opposite patterns will apply in a jetstreak’s exit region

The theory used to diagnose the circulation about fronts derives from the semi-geostrophic system of equations

Assume:

We have an east-west front

The change in the Coriolis parameter across the frontal zone can be ignored (f = constant)

The geostrophic wind in the x direction (along front) >> that the ageostrophic wind along front (ug >> uag).

In this case, the total derivative in pressure coordinates can be expressed as

py

vvx

utdt

daggg

The u momentum equation usingabsolute momentum:

xFfvxdt

du

Note that: p

fy

y

fyv

x

fyu

t

fy

dt

fydg

)()()()()( X X X

fvy

fyv

dt

fyd

)()( Add this equation to the u

Momentum equation to get:

xFxdt

fyud

dt

dm

)(

dt

d The thermodynamic equation: the rate of change of potential temperaturefollowing a parcel equals the diabatic heating rate .

yfug

1

xfvg

1 0

y

v

x

u gg

The u momentum equation

Geostrophic wind relationships

0

py

v

py

v

y

v

x

u

py

v

x

u agaggg Continuity equation

Governing equations applying geostrophic momentum approximation

xFxdt

dm

dt

d0

py

vag

u momentum equation thermodynamic equation continuity equation

Note also that we can define a streamfunction such that

and we will satisfy the continuity equation. ypvag

,

With this system of equations we seek a solution for the cross frontal circulation. To derive this, we must develop prognostic equations for the:

In the interest of time, I will derive the absolute vorticity and leave the other derivations to you…..

Absolute vorticity

The components of the cross frontal thermal wind balance

The static stability

y

m

yp

m

p

p

m

y

mv

y

mv

x

mu

t

m

ydt

dm

y aggg

y

m

pp

m

yy

m

yv

y

m

y

v

y

m

yv

y

m

y

v

y

m

xu

x

m

y

u

y

m

tdt

dm

y

ag

agg

gg

g

p

m

yy

m

y

v

y

m

y

v

x

m

y

u

y

m

dt

d

dt

dm

yaggg

xFxdt

dm

Expand d/dt operator

Take y derivative

Consolidate terms that compose d/dt (m/ y)

Momentum equation

p

m

yy

m

y

v

y

m

y

v

x

m

y

uF

xyy

m

dt

d agggx

Substitute momentum equation

and rearrange

p

m

yy

m

y

v

y

m

y

v

x

m

y

uFfv

yy

m

dt

d agggxg

Substitute

geostrophic wind to eliminate

p

fyu

yy

fyu

y

v

y

fyu

y

v

x

fyu

y

u

y

F

y

vf

y

m

dt

d ggagggggxg

p

m

yy

m

y

v

y

m

y

v

x

m

y

uFfv

yy

m

dt

d agggxg

Expand first term on

RHS and use m=ug-fy

p

u

yy

vf

y

u

y

v

y

vf

y

u

y

v

x

u

y

u

y

F

y

vf

y

m

dt

d gaggaggggggxg

p

u

yy

vf

y

u

y

v

y

F

y

v

x

u

y

u

y

vff

y

m

dt

d gaggagxgggg

Rearrange and cancel terms that add up to zero

Write individual terms

X X

p

fyu

yy

fyu

y

v

y

F

y

m

dt

d ggagx

Write remaining terms in terms of m

p

m

yy

m

y

v

y

F

y

m

dt

d agx

p

m

yy

m

py

F

y

m

dt

d x

Substitute for vag/ y from continuity eqn

p

m

yy

m

py

F

y

m

dt

d x

Vorticity Equation

p

F

y

v

p

m

y

m

p

v

p

u

y

v

p

v

y

u

p

m

dt

d xagaggggg

Other equations I will not derive:

Components inThermal wind eqn

dy

d

dp

d

p

m

pypyp

u

y

v

p

v

y

u

ydt

d gggg

ln

ppy

v

p

v

ypdt

d agag

Static stability

Changes in static stability and vorticity depend on the ageostrophic circulation

To maintain thermal wind balance, the terms on the left hand side of each of the two lower equations must be equal. We can subtract equations to get…

dy

d

p

F

p

u

y

v

p

v

y

u

dp

d

p

m

pypyy

v

p

m

y

m

p

v

xgggg

agag

2

ln

ypvag

,Now use the definition of the streamfunction to reduce this to a single equation in one unknown

dy

d

p

F

p

u

y

v

p

v

y

u

ydp

d

p

m

py

m

pyp

m

yp

xgggg

2

ln2

2

22

2

2

The equation above was originally derived by Sawyer (1956) for the special case of no along front variations in potential temperature, and modified by Eliassen (1962) to the form above.

The equation is therefore called the “Sawyer-Eliassen Equation”

dy

d

p

F

p

u

y

v

p

v

y

u

ydp

d

p

m

py

m

pyp

m

yp

xgggg

2

ln2

2

22

2

2

Right side of equation represent the forcing (known from measurements or in model solution)

Static stability Baroclinicity (thermal wind) Inertial stability

Geostrophic deformation Friction Diabatic heating

, the streamfunction, is the response

Solutions for can be obtained provided lateral and top/bottom boundary conditions are specified and the potential vorticity is positive in the domain

(air is inertially, convectively and symmetrically [slantwise] stable).

Nature of the solution of the Sawyer-Eliassen Equation:

A direct circulation (warm air rising and cold air sinking) will result with positive forcing.

An indirect circulation (warm air sinking and cold air rising) will result with negative forcing.

Cold air Warm air

Isentrope

Cold air Warm air

Isentrope

Dynamics of frontogenesis

On the figure on the left, Dashed lines: potential temperatureBlue lines: pressure surfaces (exaggerated)Shading: isotachs (blue into screen, red out)

A conceptual model of the ageostrophic circulation caused by frontogenesis

Geostrophically-balancedweak front

2. Impulsively intensify front

Stronger temperature gradientleads to more steeply sloped

pressure surfaces and an increasein the pressure gradient force

at both high and low levels

1. Initial condition

Dynamics of frontogenesisA conceptual model of the ageostrophic circulation caused by frontogenesis

2. Impulsively intensify front

Stronger temperature gradientleads to more steeply sloped

pressure surfaces and an increasein the pressure gradient force

at both high and low levels

3. Air accelerates

Air rises on warm sideAir descends on cold side

Air accelerates along isentropes toward cold air and into screen aloftAir accelerates toward warm air and

out of screen in low levels

Dynamics of frontogenesisA conceptual model of the ageostrophic circulation caused by frontogenesis

3. Air accelerates

Air rises on warm sideAir descends on cold side

Air accelerates toward cold air and into screen aloft

Air accelerates toward warm air andout of screen in low levels

Air cools at moistAdiabatic lapse rate

Air warms at dryAdiabatic lapse rate

4. Balance is restored

- Air rises and cools on warm side- Air sinks and warms on cold side- counteracts effects of frontogenesis

-Wind speed in upper jet increases (into screen)-Wind speed in lower jet increases (out of screen) - Coriolis force increases- Geostrophic balance restored

The circulation describe in the last few slides can be seen clearly on the frontillustrated on the cross section below

p

u

y

v

p

v

y

u gggg2In the absence of diabatic heating and friction, the forcing for the SW circulation can be expressed as

Using the thermal wind relationship yp

m

And the expression for the non-

divergence of the geostrophic wind0

y

v

x

u gg

This expression can be written:

xy

u

yx

u gg 2

Consider a jetstreak where =0

In the entrance quadrant ug increases with xwhile decreases with y.

x

02

xy

u

yx

u gg DIRECTCIRCULATION

02

xy

u

yx

u gg INDIRECTCIRCULATION

In the exit quadrant ug decreases with xwhile decreases with y.

p

u

y

v

p

v

y

u gggg2In the absence of diabatic heating and friction, the forcing for the SW circulation can be expressed as

Using the thermal wind relationship yp

m

And the expression for the non-

divergence of the geostrophic wind0

y

v

x

u gg

This expression can be written:

xy

u

yx

u gg 2

Consider a shear zone along a temperature gradient where =0

ug decreases with y while increases with x.

INDIRECTCIRCULATION

02

xy

u

yx

u gg

x

ug

Cold advection pattern corresponds to an indirect circulation

Warm advection pattern corresponds to an direct circulation

Correspondingly:

Example of solution of the Sawyer-Eliassen equation

The circulation about an ideal frontal zone characterized by

Confluence (top)

Shear (bottom)

Streamlines of ageostrophic circulation (thick solid lines)Isotachs of ug (denoted U) (dashed lines)Isotachs fo vg (denoted V) (thin solid lines)

IMPACT OF THERMAL ADVECTION ON JETSTREAKS

In the following diagrams: (+) means positive (downward motion) (-) means negative (upward motion) Circulation is direct if upward motion (-) is south of downward motion (+) since cold air lies to north

Jetstreak with no temperature gradient along jet axis: direct circulation in entrance, indirect in exit, symmetric around the axis of the jetstreak

Jetstreak with temperature gradient along jet axis: cold air advection maximum along jet axis. Air descends along jet axis creating two direct circulations, one on either side of of jet.

rarely occurs!

IMPACT OF THERMAL ADVECTION ON JETSTREAKS

In the following diagrams: (+) means positive (downward motion) (-) means negative (upward motion) Circulation is direct if upward motion (-) is south of downward motion (+) since cold air lies to north

Jetstreak with cold advection: direct circulation in entrance shifted to south side of jetstreak axis, indirect circulation shifted to north side of jetstreak axis.

Common when jetstreak is on west side of trough

Common when jetstreak is on east side of trough

Jetstreak with warm advection: direct circulation in entrance shifted to north side of jetstreak axis, indirect circulation shifted to south side of jetstreak axis.

Schematic illustration of tropopause folding and the development of an upper level frontal zone

Corresponding example in the real world

Warm air

DirectCell

IndirectCell

Cold air

Active mixing and exchange layer

The complete description of a jetstreak passing through a baroclinic wave must also include the effects of flow curvature

Curvature shifts the direct circulation in jet entrance region toward north side of jet axis

Curvature shifts the indirect circulation in jet exit region toward north side of jet axis

Relationship of upper level fronts to evolving baroclinic waves

1000 mb height, 500 mb height, surface frontStippling: precipitationCross hatching: 500 mb frontal zoneLines: cross sections on next figures

Front distinct Front diffuse Fronts distinct

Front distinctFront diffuse

ON CROSS SECTION

On the previous panel:

Jestreak propagates from W side of trough to E side

Frontal structure on corresponding cross sections is distinct on W side, then both sides, then E side – upper level frontogenesis is tied to the secondary circulations about the jet

Strong jetSharp front

Tilting dominantFrotogenetic process

Frontal zone advected around trough and enhanced by confluence