when advection destroys balance, vertical circulations arise

When advection destroys balance, vertical circulations arise

COMET-MSC Winter Weather Course

29 Nov. - 10 Dec. 2004

ppt started from one by

James T. Moore

Saint Louis University

Cooperative Institute for Precipitation Systems

Brian Mapes

Quasi-Geostrophic Theory

• It provides a framework to understand the evolution of balanced three-dimensional velocity fields.

• It reveals how the dual requirements of hydrostatic and geostrophic balance (encapsulated as thermal wind balance) constrain atmospheric motions.

• It helps us to understand how the balanced, geostrophic mass and momentum fields interact on the synoptic scale to create vertical circulations which result in sensible weather.

Stable balanced dynamics• Deviations from balance lead to force imbalances

that drive ageostrophic and vertical motions which adjust the state back toward balance.

• Consider hydrostatic, geostrophic as simplest case of balances.

• Houze chapter 11 - use Boussinesq, hydrostatic equation set as we did for gravity waves.

• Introduce pseudoheight

• Assume wind is mostly geostrophic ug, vg

• Note: f-plane approximation means Vg =0

Balance in atmospheric dynamics1. The vertical equation of motion: imbalance between

the 2 terms on the RHS results in small vertical motions that restore balance - unless the state is gravitationally unstable

2. The horizontal equation of motion: imbalance between the major terms on the RHS leads to small ageostrophic motions that restore balance - unless the state is inertially unstable

3. Between lies symmetric instability. Like gravitional instability, it has moist (potential, conditional) cousins. For now, STABLE CASE

( )∇ +⎛

⎝⎜

⎞

⎠⎟ =− − ⋅∇ + ∇ − ⋅∇

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

2 02 2

20 21f

P

f

PV V

Pg g gσ

∂

∂ω

σ

∂

∂η

σ

∂

∂

r r Φ

Old school: Quasi-Geostrophic Omega Equation

(vorticity-oriented form)

A B C

Term A: three-dimensional Laplacian of omega

Term B: vertical variation of the geostrophic advection of the absolute geostrophic vorticity

Term C: Laplacian of the geostrophic advection of thickness

Problems with the Traditional Form of Q-G Diagnostic Omega Equation

• The two forcing functions are NOT independent of each other

• The two forcing functions often oppose one another (e.g., PVA and cold air advection – who wins?)

• You need more than one level of information to estimate differential geostrophic vorticity advection

• You cannot estimate the Laplacian of the geostrophic thickness advection by eye!

• The forcing functions depend upon the reference frame within which they are measured (i.e., the forcing functions are NOT Galilean invariant)

PV view of how maintenance of balance requires vertical motions

cyclonic (Trof)

Thermal wind balance prevails: There is a Z trough (trof) for geostrophic balance, with a cold core beneath it, supporting it hypsometrically (in hydrostatic balance).

cyclonic(Trof)

Unsheared advection of T, u, v, vort, PV: no problem, whole structure moves

Sheared advection breaks thermal wind balance

cyclonic (Trof)

Sheared advection breaks thermal wind balance

Z Trof(hypsometric)

The PV view of balanced circulation: (Rob Rogers’s fig)

Long-lived Great Plains MCV Hurricane Andrew after landfall

Potential temperature and potential vorticity cross sections

Q-vector Form of the Q-G Diagnostic Omega Equation

Alternate approach developed by Hoskins et al. (1978, Q. J.) – manipulated the equations so forcing is 1 term, not 2:

( )

( )

∇ +⎛

⎝⎜

⎞

⎠⎟ =− ∇⋅

= =− ⋅∇ ⋅∇⎛

⎝⎜⎜

⎞

⎠⎟⎟

= =− ⋅∇ ⋅∇⎛

⎝⎜⎜

⎞

⎠⎟⎟

−

p

x y

g

p

g

p

x y

g

p

g

p

f

PQ where

Q Q QR

P

V

xT

V

yT

or

Q Q QRP

P

V

x

V

y

2 02 2

2

1

0

2σ

∂

∂ω

σ

∂

∂

∂

∂

σ

∂

∂θ

∂

∂θ

κ

κ

r

rr r

rr r

,

, ,

, ,

Q-vector Form of the Q-G Diagnostic Omega Equation

∇ +⎛

⎝⎜

⎞

⎠⎟ =− ∇⋅p

f

PQ2 0

2 2

2 2σ

∂

∂ω

r

Treat Laplacian as a “sign flip” Then,

If -2•Q > 0 (convergence of Q) then < 0 (upward vertical motion)

If -2•Q < 0 (divergence of Q) then > 0 (downward vertical motion)

The Q vector points along the ageostrophic wind in the lower branch of the secondary circulation

Q vectors point toward the rising motion and are proportional to the strength of the horizontal ageostrophic wind

Advantages of Using Q Vectors

• You only need one isobaric level to compute the total forcing (although layers are probably better to use)

• Only one forcing term, so no cancellation between terms

• Plotting Q vectors indicates where the forcing for vertical motion is located and they are a good approximation for the ageostrophic wind

• The forcing function is not dependent on the reference frame (I.e., it is Galilean invariant

• Plotting Q vectors and isentropes can indicate regions of Q-G frontogenesis/frontolysis

• No term is neglected (as in the Trenberth method which neglects the deformation term)

Interpreting Q Vectors

( )rr r

Q Q QRP

P

V

x

V

yx y

g

p

g

p= = − ⋅∇ ⋅∇⎛

⎝⎜⎜

⎞

⎠⎟⎟

−

, ,κ

κσ

∂

∂θ

∂

∂θ

1

0

Qu

x x

v

x y

and

Qu

y x

v

y y

x

g g

y

g g

=− +⎡

⎣⎢

⎤

⎦⎥

=− +⎡

⎣⎢

⎤

⎦⎥

∂∂

∂θ∂

∂∂

∂θ∂

∂∂

∂θ∂

∂∂

∂θ∂

Setting aside the coefficients,

Expanding Q and assuming adiabatic conditions yields the following expression for Q:

Qu

x x

v

x yx

g g=− +

⎡

⎣⎢

⎤

⎦⎥

∂∂

∂θ∂

∂∂

∂θ∂

Interpretation of Qx

cold warm

θ

ug

θ

cold warm

Geostrophic stretching

deformation weakens θ

θ

cold

warm

Geostrophic shearing

deformation turns θ

vg

θ

cold

warm

to

to+t

Qu

y x

v

y yy

g g=− +

⎡

⎣⎢

⎤

⎦⎥

∂∂

∂θ∂

∂∂

∂θ∂

Interpretation of Qy

cold warm

θ

ug

θ

cold

warm

Geostrophic shearing

deformation turns θ

θ

cold

warm

Geostrophic stretching

deformation strengthens θ

vg

θ

cold

warm

to

to+t

Keyser et al. (1992, MWR) derived a form of the Q vector in “natural” coordinates where one component is oriented parallel to isotherms and another component is oriented normal to the isotherms.

In this form one component (Qs) has the two shearing deformation terms, expressing rotation of isotherms, that normally show up in Qx and Qy . Meanwhile, the other component (Qn) has the two stretching deformation terms expressing the contraction or expansion of isotherms.

We will see that this novel form of the Q vector has distinct advantages, in terms of interpretation.

An Alternative form of Q in “natural” coordinates

Defining the Orientation of Qs and Qn with Respect

to θ

Martin (1999, MWR)

Qs is the component of Q associated with rotating the thermal gradient.

Qn is the component of Q associated with changing the magnitude of the thermal gradient.

θ

θ-1

θ+1

θ+2

θ

Qs

Qn Qcold

warm

s

n

Keyser et al. (1992, MWR)

Qx

V

x y

V

y

whenx

Qy

V

y

Qy

u

yi

v

yj

xi

yj

Qy

v

n

g g

n

g

n

g g

n

=∇

• ∇⎛

⎝⎜⎜

⎞

⎠⎟⎟ + • ∇

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

=

=∇

• ∇⎛

⎝⎜⎜

⎞

⎠⎟⎟

=∇

+⎛

⎝⎜

⎞

⎠⎟ • +

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

=∇

1

0

1

1

1

θ∂θ∂

∂∂

θ∂θ∂

∂∂

θ

∂θ∂

θ∂θ∂

∂∂

θ

θ∂θ∂

∂∂

∂∂

∂θ∂

∂θ∂

θ∂θ∂

∂

r r

r;

$ $ $ $

g

y y∂∂θ∂

⎛

⎝⎜

⎞

⎠⎟

Defining Qn and Interpreting What It Means

Qy

v

y yn

g=

∇⎛

⎝⎜

⎞

⎠⎟

1θ

∂θ∂

∂∂

∂θ∂

Defining Qn and Interpreting What It Means (cont.)

θθ+1 θ+2

vg/y < 0; therefore Qn <0;

Qn points from cold to warm air; confluence (diffluence) in wind field implies frontogenesis (frontolysis)

Qnθ

Couplets of div Qn:

• Tend to line up across the isotherms

• Show the ageostrophic response to the geostrophically-forced packing/unpacking of

the isotherms

• Often exhibit narrow banded structures typical of the “frontal” scale

• Give an indication of how “active” a front might be

Interpreting Q vectors: Qn

Advection by geostrophic stretching deformation acts to change the magnitude of the thermal gradient vector, θ.

But the same geostrophic advection changes the wind shear in the direction OPPOSITE to that needed to restore balance. This is why the forcing for ageostrophic secondary circ is -2x(.Q)!

cold

Low level wind: pure geostrophic deformation (noting .Vg = 0), here acting to weaken dT/dx.

warm

Thermal wind

Upper level wind: Upper level wind: addadd thermal windthermal wind toto low levellow level windwind. . v component is positive v component is positive and decreases to north, so advection is and decreases to north, so advection is acting to acting to increaseincrease upper-level v upper-level v..

Qy

V

x x

V

y

whenx

Qy

V

x

Qy

u

xi

v

xj

xi

yj

Qy

v

s

g g

s

g

s

g g

s

=∇

• ∇⎛

⎝⎜⎜

⎞

⎠⎟⎟ − • ∇

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

=

=∇

• ∇⎛

⎝⎜⎜

⎞

⎠⎟⎟

=∇

+⎛

⎝⎜

⎞

⎠⎟ • +

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

=∇

1

0

1

1

1

θ∂θ∂

∂∂

θ∂θ∂

∂∂

θ

∂θ∂

θ∂θ∂

∂∂

θ

θ∂θ∂

∂∂

∂∂

∂θ∂

∂θ∂

θ∂θ∂

∂

r r

r;

$ $ $ $

g

x y∂∂θ∂

⎛

⎝⎜

⎞

⎠⎟

Defining Qs and Interpreting What It Means

Defining Qs and Interpreting What It Means (cont.)

Qy

v

x ys

g=

∇⎛

⎝⎜

⎞

⎠⎟

1θ

∂θ∂

∂∂

∂θ∂

θ

θ+1

θ+2θ

vg/x > 0; therefore Qs > 0.

Qs has cold air is to its left, causes cyclonic rotation of the vector θ. Thermal wind balance thus requires v to increase aloft, but geostrophic advection acts to decrease v aloft.

QsQs

Thermal wind

Upper wind Couplets of div Qs:

• Tend to line up along the isotherms

• Show the ageostrophic response to the geostrophically-forced turning of the

isotherms

• Tend to be oriented upstream and downstream of troughs

• Are associated with the synoptic wave scale of ascent and descent

Estimating Q vectors

Sanders and Hoskins (1990, WAF) derived a form of the Q vector which could be used when looking at weather maps to qualitatively estimate its direction and magnitude:

rr

QR

P

T

yk

V

xg

=− ×⎡

⎣⎢⎢

⎤

⎦⎥⎥

∂∂

∂∂

$

Where the x axis is defined to be along the isotherms (with cold air to the left) and y is normal to x and to the left.

Thus, Q is large when the temperature gradient is strong and when the geostrophic shear along the isotherms is strong.

To estimate the direction of Q just use vector subtraction to compute the derivative of Vg along the isotherms, then rotate the vector by 90° in the clockwise direction. Example:

AB

AB

Holton (1992)

A - B

=

Q

A-

B

=

Q

Jet Entrance Region

Col Region

90 deg

90 deg

Q vectors

This is mainly the cross-front, n

component Qn

Q vectors in a setting where warm air rises

Qn vectors

Direct Thermal CirculationConfluent Flow

Holton, 1992

cold

warm

Vageo

North South

VageoThermally Indirect Circulation

QJet Exit Region

Q vectors in a setting where COLD air rises

Holton (1992)

Idealized pattern of sea-level isobars (solid) and isotherms (dashed) for a train of cyclones and anticyclones. Heavy bold arrows are Q vectors. This is mostly the along-front or s

component Qs.

Semi-geostrophic extension to QG theory

• Allow advection of b and v by an ageostrophic horizontal wind ua in cross-front (x) direction only (following Houze section 11.2.2).

•An elegant trick: define

•Using the fact that Dvg/Dt = -fua, the total derivative in X space

becomes analogous to Dg/Dt:

Semi-geostrophic extension to QG theory (cont)

• More elegant trickery:

•Defining the geostrophic PV (Houze 11.50)

One can get the streamfunction equation (11.60)

Comparing the QG case (11.20)

•PV plays the role of a static stability in this system.

Another form (from notes of R. Johnson, CSU)

is met (translation: PV must be positive, so that the system is symmetrically stable)

Frontogenesis (definition)

θpDt

DF ∇=

The 2-D scalar frontogenesis function (F ):

F > 0 frontogenesis, F < 0 frontolysis

(S. Petterssen 1936)

F: generalization of the quasi-geostrophic version, the Q-vector

Can also include diabatic heating gradients, etc.

θpDt

DF ∇=Q

g

Frontogenesis and Symmetric Instability

Symmetric instabilities, contributing to banded

precipitation, often north and east of midlatitude cyclones

Mesoscale Instabilities and Processes Which Can Result in Enhanced Precipitation

• Conditional Instability• Convective Instability• Inertial Instability• Potential Symmetric Instability• Conditional Symmetric Instability• Weak Symmetric Stability• Convective-Symmetric Instability• Frontogenesis

Balance in atmospheric dynamics1. The vertical equation of motion: imbalance between

the 2 terms on the RHS results in small vertical motions that restore balance - unless the state is gravitationally unstable

2. The horizontal equation of motion: imbalance between the major terms on the RHS leads to small ageostrophic motions that restore balance - unless the state is inertially unstable

3. Between lies symmetric instability. Like gravitional instability, it has moist (potential, conditional) cousins. For now, STABLE CASE

Schultz et al. 1999 MWR

Instabilities: nomenclatureSchultz et al. MWR 1999

“The intricacies of instabilities”

Conditional Symmetric Instability: Cross section of θes and Mg taken normal to the 850-300 mb thickness contours

θes

Mg +1

θes+ 1θes-1

Mg

Mg -1

s

Note: isentropes of θes

are sloped more verticalthan lines of absolutegeostropic momentum,Mg.

Vert.stableHoriz.

stable

Symm.Symm.unstableunstable

Conditional Symmetric Instability in the Presence of Synoptic Scale Lift – Slantwise Ascent and Descent

Multiple Bands with Slantwise Ascent

Frontogenesis and varying Symmetric Stability

• Emanuel (1985, JAS) has shown that in the presence of weak symmetric stability (simulating condensation) in the rising branch, the ageostrophic circulations in response to frontogenesis are changed.

• The upward branch becomes contracted and becomes stronger. The strong updraft is located ahead of the region of maximum geostrophic frontogenetical forcing.

• The distance between the front and the updraft is typically on the order of 50-200 km

• On the cold side of the frontogenetical forcing stability is greater and and the downward motion is broader and weaker than the updraft.

Frontal secondary circulation - constant stability

Frontal secondary circ - with condensation on ascent

Emanuel (1985, JAS)

Schematic of Convective-Symmetric Instability Circulation

Blanchard, Cotton, and Brown, 1998 (MWR)

Convective-Symmetric Instability

Multiple Erect Towers with Slantwise Descent

Sanders and Bosart, 1985: Mesoscale Structure in the Megalopolitan Snowstorm of 11-12 February 1983. J. Atmos. Sci., 42, 1050-1061.

Frontogenesis and Symmetric Instability

NW

SE

NW-SE cross-section shown on next slide.

A Conceptual Model: Plan View of Key Processes

Often found in the vicinity of an extratropical cyclone warm front, ahead of a long-wave trough in a region of strong, moist, mid-tropospheric southwesterly flow

Dry Air

es

Convectively Unstable

Shaded area = CSI

CSI

Arrows = Ascent zone F = Frontogenesis zone

Heavy snow area

A Conceptual Model: Cross-Sectional View of Key Processes

CSI may be a precursor to elevated CI, as the vertical circulation associated with CSI may overturn e surfaces with time creating convectively unstable zones aloft

Nolan-Moore Conceptual Model

• Many heavy precipitation events display different types of mesoscale instabilities including:– Convective Instability (CI; θe decreasing with

height)– Conditional Symmetric Instability (CSI; lines

of θes are more vertical than lines of constant absolute geostrophic momentum or Mg)

– Weak Symmetric Stability (WSS; lines of θes

are nearly parallel to lines of constant absolute geostrophic momentum or Mg)

Spectrum of Mesoscale Instabilities


• These mesoscale instabilities tend to develop from north to south in the presence of strong uni-directional wind shear (typically from the SW)

• CI tends to be in the warmer air to the south of the cyclone while CSI and WSS tend to develop further north in the presence of a cold, stable boundary layer.

• It is not unusual to see CI move north and become elevated, producing thundersnow.


• CSI may be a precursor to elevated CI, as the vertical circulation associated with CSI may overturn θe surfaces with time creating convectively unstable zones aloft.

• We believe that most thundersnow events are associated with elevated convective instability (as opposed to CSI).

• CSI can generate vertical motions on the order of 1-3 m s-1 while elevated CI can generate vertical motions on the order of 10 m s-1 which are more likely to create charge separation and lightning.

Parting Thoughts on Banded Precipitation (Jim Moore)

• Numerical experiments suggest that weak positive symmetric stability (WSS) in the warm air in the presence of frontogenesis leads to a single band of ascent that narrows as the symmetric stability approaches neutrality.

• Also, if the forcing becomes horizontally widespread and EPV < 0, multiple bands become embedded within the large scale circulation; as the EPV decreases the multiple bands become more intense and more widely spaced.

• However, more research needs to be done to better understand how bands form in the presence of frontogenesis and CSI.

Figure from Nicosia and Grumm (1999,WAF). Potential symmetric instability occurs where the mid-level dry tongue jet overlays the low-level easterly jet (or cold conveyor belt), north of the surface low. In this area dry air at mid-levels overruns moisture-laden low-level easterly flow, thereby steepening the slope of the θe surfaces.

Nicosia and Grumm (1999, WAF) Conceptual Model for CSI

• Also….since the vertical wind shear is increasing with time the Mg surfaces become more horizontal (become flatter). Thus, a region of PSI/CSI develops where the surfaces of θe or θes are more vertical than the Mg surfaces.

• In this way frontogenesis and the development of PSI/ CSI are linked.

Frontogenesis (definition)

θpDt

DF ∇=

The 2-D scalar frontogenesis function (F ):

F > 0 frontogenesis, F < 0 frontolysis

(S. Petterssen 1936)

F: generalization of the quasi-geostrophic version, the Q-vector

Can also include diabatic heating gradients, etc.

θpDt

DF ∇=Q

g

Vector Frontogenesis Function

(Keyser et al. 1988, 1992)

snF sn FF +=

)( θ

θ

pDtD

knsF

pDtD

nF

∇×⋅=

∇−=

Change in magnitude

Corresponds to vertical motion on the frontal scale (mesoscale bands), as cross-frontal F vector points along low-level Va, toward upward motion.

Change in direction (rotation)

Corresponds to vertical motion on the scale of the baroclinic wave itself: rotation of T gradient by a cyclone’s winds causes along-front F vectors to converge on east side of low pressure

Three-Dimensional Frontogenesis Equation

Terms 1, 5, 9: Diabatic Terms

Terms 2, 3, 6, 7: Horizontal Deformation Terms

Terms 10 and 11: Vertical Deformation Terms

Terms 4 and 8: Tilting Terms

Term 12: Vertical Divergence Terms

Bluestein (Synoptic-Dynamic Met. In Mid-Latitudes, vol. II, 1993)

1 2 3 4

5 6 7 8

9 10 11 12

Assumptions to Simplify the Three-Dimensional Frontogenesis Equation

θ

θ+ 1

θ+ 2

y’

x’

• y’ axis is set normal to the frontal zone, with y’ increasing towards the cold air (note: y’ might not always be normal to the isentropes)

• x’ axis is parallel to the frontal zone

• Neglect vertical and horizontal diffusion effects

Fd

dt y

u

y x

v

y y

w

y z y

d

dt=

−′

⎛⎝⎜

⎞⎠⎟ =

′ ′+

′ ′+

′−

′⎛⎝⎜

⎞⎠⎟

∂θ∂

∂∂

∂θ∂

∂∂

∂θ∂

∂∂

∂θ∂

∂∂

θ

Simplified Form of the Frontogenesis Equation

A B C D

Term A: Shear term

Term B: Confluence term

Term C: Tilting term

Term D: Diabatic Heating/Cooling term

Frontogenesis: Shear Term

Shearing Advection changes orientation of isotherms

Carlson, 1991 Mid-Latitude Weather Systems

Frontogenesis: Confluence Term

Cold advection to the north

Warm advection to the south


Carlson (Mid-latitude Weather Systems, 1991)

Shear and Confluence Terms near Cold and Warm Fronts

Shear and confluence terms oppose one another near warm fronts

Shear and confluence terms tend to work together near cold fronts

Frontogenesis: Tilting Term

Adiabatic cooling to north and warming to south increases horizontal thermal gradient


Frontogenesis: Diabatic Heating/Cooling Term

frontogenesis

frontolysis

T constant T increases

T increases T constant


Petterssen (1968)

Frontogenesis/Frontolysis with Deformation with No Diabatic Effects or Tilting Effects

{ }Fd

dtDef DivR= ∇ = ∇ −θ θ β

1

22cos

Defv

x

u

y

u

x

v

yR = +⎛⎝⎜

⎞⎠⎟ + −

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∂∂

∂∂

∂∂

∂∂

2 212

= angle between the isentropes and the axis of dilatation

where:

and

Kinematic Components of the Wind

Translation

Divergence

Vorticity

Deformation

Stretching and Shearing Deformation Patterns

Stretching

Deformation

Shearing

Deformation

Stretching Deformation Patterns

Bluestein (1992, Synoptic-Dynamic Met)

Stretching along the flow

Stretching normal to the flow

Translational component of wind removed


Shearing Deformation Patterns

Bluestein (1992, Synoptic-Dynamic Met)

Stretching in a direction 45° to the left of the flow

Stretching in a direction 45° to the right of the flow



Petterssen (Weather Analysis and Forecasting, vol. 1, 1956)

< 45°

> 45°

Axis of dilatation

Axis of dilatation

Frontogenesis

Frontolysis

Pure Deformation Wind Field Acting on a Thermal Gradient

Keyser et al. (MWR, 1988)

Isotherms are rotated and brought closer together

Deficiencies of Kinematic Frontogenesis

• Fronts can double their intensity in a matter of several hours; kinematic frontogenesis suggests that it takes on the order of a day.

• Kinematic frontogenesis does not account for changes in the divergence of momentum fields; values of divergence and vorticity in frontal zones are on scales <= 100 km, suggesting highly ageostrophic flow.

• Kinematic frontogenesis fails since temperature is treated as a passive scalar. As the thermal gradient changes the thermal wind balance is upset, therefore there is a continual readjustment of the winds in the vertical in an attempt to re-establish geostrophic balance.

Carlson (Mid-Latitude Weather Systems, 1991)

Frontogenetical Circulation

• As the thermal gradient strengthens the geostrophic wind aloft and below must respond to maintain balance with the thermal wind.

• Winds aloft increase and “cut” to the north while winds below decrease and “cut” to the south, thereby creating regions of div/con.

• By mass continuity upward motion develops to the south and downward motion to the north – a direct thermal circulation.

• This direct thermal circulation acts to weaken the frontal zone with time and works against the original geostrophic frontogenesis.

West East

West East

Ageostrophic Adjustments in Response to Frontogenetical Forcing

North South

Thermally Direct Circulation

Strength and Depth of the vertical circulation is modulated by static stabilityCarlson (Mid-latitude Weather Systems, 1991)

Frontogenetical Circulation

WARMCOLD

Sawyer-Eliassen Description of the Frontogenetic Circulation

• Includes advections by the ageostrophic component of the wind normal to the frontal zone or jet streak.

• The ageostrophic and vertical components of the wind are viewed as nearly instantaneous responses to the geostrophic advection of temperature and geostrophic deformation near the frontal zone.

• The cross-frontal (transverse) ageostrophic component of the tranverse/vertical circulations is significant and can become as large in magnitude as the geostrophic wind velocity.

• Thus, divergence/convergence and vorticity production in the vicinity of the front take place more rapidly than predicted by purely kinematic frontogenesis.

Carlson (Mid-latitude Weather Systems,1991)

Frontogenetical Circulation Factors

According to the Sawyer-Eliassen equations (see Carlson, Mid-Latitude Weather Systems, 1991):

• The major and minor axes of the elliptical circulation are determined by the relative magnitudes of the static stability and the absolute geostrophic vorticity; the vertical slope is a function of the baroclinicity.

• High static stability compresses and weakens the circulation cells.

• If the absolute geostrophic vorticity is small (weak inertial stability) in the presence of high static stability the circulation ellipses are oriented horizontally.

• If the absolute geostrophic vorticity is large (strong inertial stability) in the presence of small static stability the circulation cells are oriented vertically.

• High static stability and low inertial stability

Result is a shallow but broad circulation.

With high static stability, a little vertical motion results in large change in temperature.

With low inertial stability, takes longer for Coriolis force to balance the pressure gradient force.

Greg Mann, 2004

• Low static stability and high inertial stability

With low static stability, need large vertical motion to change the temperature.

With high inertial stability, Coriolis force quickly balances the pressure gradient force.

Greg Mann, 2004

Role of symmetric stability

• Symmetric stability plays a large role in determining the strength and width of the ageostrophic frontal circulation– Small symmetric stability

• Intense and narrow updraft

– Large symmetric stability• Broad and weak updraft.

Greg Mann, 2004

( )

[ ]

[ ]

Fd

dt

F F n F s

F div Def

F Def

n s

n R

s R

= ∇

= +

= ∇ −

= ∇ +

r

r

r

r

θ

θ β

θ ζ β

$ $

cos( )

sin( )

12

2

12

2

Defining Fs and Fn Vectors from the Frontogenesis Function

Keyser et al. (1988, MWR)

Defining Fs and Fn Vectors from the Frontogenesis Function

Keyser et al. (1988, MWR) and Augustine and Caracena (1994, WAF)

Interpreting F Vectors

• The component of F normal to the isentropes (Fn) is the frontogenetic component; it is equivalent but opposite in sign to the Petterssen frontogenesis function. When F is directed from cold to warm (Fn < 0), the local forcing is frontogenetic, i.e., the large scale is acting to fortify the frontal boundary by strengthening the horizontal potential temperature gradient and increasing the slope of the isentropes.

• Conversely, when F is directed from warm to cold (Fn > 0), the forcing is acting in a frontolytic fashion.

• The component of F parallel to the isentropes (Fs) quantifies how the forcing acts to rotate the potential temperature gradient.

• The F vector is equivalent to the Q vector only when the horizontal wind is geostrophic; thus F is less restrictive. The divergence of F is a only a good approximation of the Q-G forcing for vertical motion when the wind is in approximate geostrophic balance.

• However, F vector convergence does NOT necessarily imply upward vertical motion.

)

Augustine and Caracena (1994, WAF)

Application of Frontogenetical Vectors for MCS Formation

Synoptic setting favorable for large MCS development.

Dashed lines are isentropes and arrows are F vectors, at 850 hPa. Red arrow indicates the low-level jet.