Advanced SynopticM. D. Eastin Frontogenesis – Kinematics & Dynamics

Advanced Synoptic M. D. Eastin

Frontogenesis – Kinematics & Dynamics


Frontal Evolution: An Example

Kinematic Frontogenesis

• Three-Dimensional (3D) Frontogenesis• Two-Dimensional (2D) Frontogenesis• Deficiencies and Limitations

Dynamic Frontogenesis

• Review of QG Theory• Semi-geostrophic (SG) Theory• Conceptual Model• Impact of Ageostrophic Advection• Application of Q-vectors to Frontogenesis

Frontogenesis – Kinematics & Dynamics


Frontal EvolutionAn Example from Observations:

Boulder Tower Observations11-12 December 1975

Time-Height Cross-Section


An Example from Observations:

• Notice how the temperature gradient strengthens between 1200 and 0000 GMT• How does this strengthening occur (and so fast)?

Frontal Evolution

From Shapiro et al

(1985)


Definitions and Our Approach:

• Intensification → Frontogenesis• Weakening → Frontolysis

• The traditional measure of frontogenesis was introduced by Petterssen (1936) to explore the kinematic processes that influence the strength of the potential temperature (θ) gradient as a function of time – called the Frontogenetic Function (F)

F > 0 → Frontogenesis

F < 0 → Frontolysis

• We shall first examine the kinematic effects whereby advection, shear, and local heating act to increase the density gradient

• Then, we will examine the dynamic effects whereby forces induced as a result of the kinematic changes produce circulations that can enhance the kinematic effects

Dt

DF



Three-dimensional (3D):

• If we expand total derivative applied to F using the thermodynamic equation – after much math – we arrive at:

• Which of these terms are “significant”? → Perform scale analysis

• Simply with a different coordinate system? → Transform to “front-normal”


x

w

zx

v

yx

u

xdt

d

xp

p

cxF

p

011

y

w

zy

v

yy

u

xdt

d

yp

p

cy p

01

z

w

zz

v

yz

u

xdt

dp

zc

p

z p

0

Diabatic

Horizontal Deformation

Vertical Deformation

Tilting

Vertical Divergence

Weighting Factors = Magnitude of θ-gradient in one direction Magnitude of the total 3D θ-gradient


Two-dimensional (2D): In a “front-relative” coordinate system

• If we define our coordinate system so that our x’-axis is parallel to the front, and our y’-axis is perpendicular (or normal) to the front, then we can simply the 3D equation

[Equation 6.2 in Lackmann text]

Note: This equation describes frontogenesis in a Lagrangian sense (following the flow)

Thus, it will NOT indicate whether the overall front is intensifying → only along small sections of the front


x’y’

yypy

v

yy

u

xF

Shearing TiltingConfluence Diabatic

Note: The “front-relative” wind components become

x’ → u’y’ → v’


Shearing Frontogenesis: In a “front-relative” coordinate system

• Describes the change in frontal strength due to differential potential temperature advection by the front-parallel (x’) wind component (u’)

• Stronger forcing near the surface


yypy

v

yy

u

xF


Initial Time Later Time


Shearing Frontolysis: In a “front-relative” coordinate system

• Describes the change in frontal strength due to differential potential temperature advection by the front-parallel (x’) wind component (u’)



yypy

v

yy

u

xF




Confluence Frontogenesis: In a “front-relative” coordinate system

• Describes the change in frontal strength due to potential temperature advection by the front-normal (y’) wind component (v’)

• Strongest forcing near the surface


yypy

v

yy

u

xF




Tilting Frontogenesis: In a “front-relative” coordinate system

• Describes the change in frontal strength due to differential potential temperature advection by vertical motion (ω) gradients in the front-normal (y’) direction

• Weak forcing at the surface (ω ~ 0)• Strongest forcing aloft (ω larger)


yypy

v

yy

u

xF




Diabatic Frontogenesis: In a “front-relative” coordinate system

• Describes the change in frontal strength due to differential diabatic forcing on the potential temperature field


• Processes: Radiation Surface Fluxes / Surface Properties Latent Heating / Evaporational Cooling


yypy

v

yy

u

xF



Diabatic Forcing: Can be important!!!

• Notice how the equivalent potential temperature (θe) gradient behind the surface cold front changes significantly as the front passes over the Gulf Stream (upward heat and moisture fluxes)


A

B C



Equivalent Potential Temperature (θe)Surface Pressure

3D Frontogenetic Function (F)Surface Pressure

Regions we should expectfrontal intensification

and strong lift


Limitations and Deficiencies:

Potential temperature is treated as a passive scalar that is simply advected around by the geostrophic wind field (kinematics)

• Recall that QG theory assumes the flow is in hydrostatic and geostrophic balance (i.e., thermal wind balance) at all times

If we change the potential temperature field (or its gradient), should we not expect a similar change in the wind field (a dynamic response) that would be required maintain the thermal wind balance?

Fronts are observed to double their intensity within several hours, but kinematic frontogenesis suggests that it should take a day or more

Does the dynamic response to any initial kinematic changes to the potential temperature field further accelerate the frontogenesis?



Review of QG Theory:

• We learned that geostrophic advection can disrupt thermal wind balance

• Ageostrophic flow (horizontal & vertical) come about in to restore the balance

Application to Frontogenesis:

Any air parcels entering a frontal zone should experience a rapid change in temperature gradient and thermal wind balance disruption

(QG Theory? → Not so fast!)

Recall: QG theory assume small Ro

“along-front” → L ~1000 km → Ro « 1 “cross-front” → L <100 km → Ro ~ 1


L-En

R-En

R-EnL-En

LfURo /


Semi-Geostrophic (SG) Theory:

A modified version of QG theory specifically developed to address frontal circulations

Assumptions:

• Cartesian coordinates (x/y/z and u/v/w)• Boussinesq approximation (see text)• Front-relative coordinate system

along-front → x’ and u’ cross-front → y’ and v’

• Along-front flow → geostrophic (ug’)

Cross-front flow → total (vg’ + vag’)

Ageostrophic advection in the cross front directions can also modify the temperature and momentum fields

• Cross-front thermal gradient is in thermal wind balance with the where along-front geostrophic flow


x’y’

y

b

z

uf g

g

b


Semi-Geostrophic (SG) Theory:

The full set of SG equations (see Section 6.3.1 in your text) can be combined to form a single diagnostic equation (called the Sawyer-Eliassen equation) that describes how geostrophic flow may disrupt thermal wind balance near a front, and the cross-front ageostrophic circulation works to restore balance.

where: and [Equation 6.16 in Lackmann text]


g

b

222 Qz

v

yy

b

z

v

y

uff

y

w

z

b

z

g agagg

GeostrophicFlow

Cross-front AgeostrophicCirculation

y

v

yy

u

xp

RQ gg

2Cross-front

Q-vector


Conceptual Model: Frontogenesis

• Assume the low-level geostrophic flow (red vectors) is acting to concentrate the background thermal gradient (kinematic frontogenesis) → disturbs thermal wind balance

Note: The resulting low-level Q-vectors (black vectors / dots) point toward the “warm side” of the frontal zone

A. To restore balance, an ageostrophic cross-front circulation that (1) cools the warm air via expansion / ascent and (2) warms the cold air via compression / descent must develop

Note: As the thermal gradient intensifies, so does the Q-vector magnitude (enhancing Q-convergence and the cross-front circulation…)


Initial Time Later Time “Cross-Front”Cross-section

A

B

BA

1 2Q2Q2 Q2Q2

Q2


Conceptual Model: Frontogenesis

• Assume the low-level geostrophic flow is acting to concentrate the background thermal gradient (via kinematic frontogenesis) → disturbs thermal wind balance

B. To restore balance, the Coriolis torque acting on the “down-gradient” cross-front ageostrophic flow will enhance the along-front geostrophic flow, which increases the along-front vertical shear, bringing the frontal zone back toward balance


Intensification of the thermal gradientenhances the cross-front pressure gradientproducing down-gradient cross-front flow

[enhances the cross-front circulation]

Coriolis torque turns the opposing down-gradientcross-front flow into opposing along-front flow

[enhances the along-front vertical shear]


Conceptual Model: Example Case


N

N

S

S

1000-mb Isentropes1000-mb Wind Barbs

1000-mb Frontogenesis850-mb Q-vectors

Cold

Cold

Cold

Warm

Warm

850-mb Omega (ω)850-mb Q-vectors

Isentropes and Omega (ω)


Impact of Ageostrophic Advection: Rapid Frontogenesis

Feedback Loop: As the thermal gradient intensifies, so does the Q-vector magnitude and the cross-front pressure gradient, enhancing both the Q-vector

convergence and the cross-front circulation…

Since the cross-front flow (which also intensifies the thermal gradient)is a combination of geostrophic advection and ageostrophic advection,the ageostrophic advection works to both restore thermal wind balanceand simultaneously enhance the thermal gradient

With no additional mechanism to offset the effects of ageostrophicadvection → rapid near-surface frontogenesis can occur!


Q2Q2


Application of Q-Vectors:

The orientation of low-level Q-vectors to the low-level potential temperature gradient provides any easy method to infer frontogenesis or frontolysis from real-time data

• If the Q-vectors point toward warm air and cross the potential temperature gradient, then ageostrophic flow will produce frontogenesis

• If the Q-vectors point toward cold air and cross the potential temperature gradient, then ageostrophic flow will produce frontolysis

• If the Q-vectors point along the temperature gradient, then ageostrophic flow will have no impact on the temperature gradient and the frontal intensity will be steady-state

Q-vectors and Frontogenesis

Q-vectors

Example:

Note: The regions of expected and observed frontogenesis / frontolysis generally agree Part of the observed evolution is due to system motion and diabatic effects


Q-vectors and Frontogenesis

925-mb Q-vectors and Isentropes11 November 2012 at 12 UTC

925-mb Isentropes12 November 2012 at 00 UTC


ReferencesBluestein, H. B, 1993: Synoptic-Dynamic Meteorology in Midlatitudes. Volume I: Principles of Kinematics and Dynamics.

Oxford University Press, New York, 431 pp.

Bluestein, H. B, 1993: Synoptic-Dynamic Meteorology in Midlatitudes. Volume II: Observations and Theory of WeatherSystems. Oxford University Press, New York, 594 pp.

Keyser, D., M. J. Reeder, and R. J. Reed, 1988: A generalization of Pettersen’s frontogenesis function and its relation tothe forcing of vertical motion. Mon. Wea. Rev., 116, 762-780.

Lackmann, G., 2011: Mid-latitude Synoptic Meteorology – Dynamics, Analysis and Forecasting, AMS, 343 pp.

Schultz, D. M., D. Keyser, and L. F. Bosart, 1998: The effect of large-scale flow on low-level frontal structure and evolutionin midlatitude cyclones. Mon. Wea. Rev., 126, 1767-1791.

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Advanced SynopticM. D. Eastin Frontogenesis – Kinematics & Dynamics