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
Advanced Synoptic
M. D. Eastin
Frontal Evolution
An Example from Observations:
Boulder Tower Observations
11-12 December 1975
Advanced Synoptic
Time-Height Cross-Section
M. D. Eastin
Frontal Evolution
An Example from Observations:
• Notice how the temperature gradient strengthens between 1200 and 0000 GMT
• How does this strengthening occur (and so fast)?
From
Shapiro et al
(1985)
Advanced Synoptic
M. D. Eastin
Kinematic Frontogenesis
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 
D

Dt
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
Advanced Synoptic
M. D. Eastin
Kinematic Frontogenesis
Three-dimensional (3D):
• If we expand total derivative applied to F using the thermodynamic equation – after
much math – we arrive at:
1
F



 1  p0     d    u    v    w 

  
  

       
c
p
x dt
x x   y x   z x 

 p      


  1  p0     d    u    v    w 
  
  


       
y  c p  p   y  dt   x y   y y   z y 



  p0      d    u    v    w 
  




    p
z  c p   z 
dt   x z   y z   z z 

x
Diabatic
Horizontal Deformation
Vertical Deformation
Tilting
Vertical Divergence
Weighting Factors = Magnitude of θ-gradient in one direction
Magnitude of the total 3D θ-gradient
• Which of these terms are “significant”?
→ Perform scale analysis
• Simply with a different coordinate system?
→ Transform to “front-normal”
Advanced Synoptic
M. D. Eastin
Kinematic Frontogenesis
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
 u  v   
F



x y y y p y y
Shearing
Confluence
Tilting
y’
x’
Diabatic
[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
Advanced Synoptic
Note: The “front-relative” wind
components become
x’ → u’
y’ → v’
M. D. Eastin
Kinematic Frontogenesis
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
Initial Time
Advanced Synoptic
 u  v   
F



x y y y p y y
Shearing
Confluence
Tilting
Diabatic
Later Time
M. D. Eastin
Kinematic Frontogenesis
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’)
• Stronger forcing near the surface
Initial Time
Advanced Synoptic
 u  v   
F



x y y y p y y
Shearing
Confluence
Tilting
Diabatic
Later Time
M. D. Eastin
Kinematic Frontogenesis
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
Initial Time
Advanced Synoptic
 u  v   
F



x y y y p y y
Shearing
Confluence
Tilting
Diabatic
Later Time
M. D. Eastin
Kinematic Frontogenesis
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)
Initial Time
Advanced Synoptic
 u  v   
F



x y y y p y y
Shearing
Confluence
Tilting
Diabatic
Later Time
M. D. Eastin
Kinematic Frontogenesis
Diabatic Frontogenesis: In a “front-relative” coordinate system
• Describes the change in frontal strength
due to differential diabatic forcing on the
potential temperature field
 u  v   
F



x y y y p y y
• Stronger forcing near the surface
Shearing
Confluence
Tilting
Diabatic
• Processes: Radiation
Surface Fluxes / Surface Properties
Latent Heating / Evaporational Cooling
Advanced Synoptic
M. D. Eastin
Kinematic Frontogenesis
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)
B
Advanced Synoptic
A
C
M. D. Eastin
Kinematic Frontogenesis
Two-Dimensional (2D): Analysis Example
• Many software packages can compute / plot the 3D or 2D frontogenetic function
• Can be useful for weather forecasting → Identify and track frontal locations
→ Anticipate differential frontal motions
→ Identify regions of strongest forcing
(correspond to regions of strong lift)
Equivalent Potential Temperature (θe)
Surface Pressure
3D Frontogenetic Function (F)
Surface Pressure
Regions we should expect
frontal intensification
and strong lift
Advanced Synoptic
M. D. Eastin
Kinematic Frontogenesis
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?
Advanced Synoptic
M. D. Eastin
Dynamic Frontogenesis
Review of QG Theory:
• We learned that geostrophic advection
can disrupt thermal wind balance
L-En
• 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!)
R-En
L-En
R-En
Recall: QG theory assume small Ro
Ro  U / f L
“along-front” → L ~1000 km → Ro « 1
“cross-front” → L <100 km → Ro ~ 1
Advanced Synoptic
M. D. Eastin
Dynamic Frontogenesis
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’
y’
• Along-front flow → geostrophic (ug’)
 Cross-front flow → total (vg’ + vag’)
x’
 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
along-front geostrophic flow
Advanced Synoptic
b where b  g 
f


z
y
ug
M. D. Eastin
Dynamic Frontogenesis
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.
 g 
b   w 

  z
z   y 


 
u g   vag

f f 


y   z 



 b     vag

2
  y  z


y



b
g 
and

2Q2
Geostrophic
Flow
Cross-front Ageostrophic
Circulation
where:



[Equation 6.16 in Lackmann text]

Q2  
Advanced Synoptic
R    ug   vg 



p  x y y y 
Cross-front
Q-vector
M. D. Eastin
Dynamic Frontogenesis
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
“Cross-Front”
Cross-section
Later Time
B
Q2
Q2
Q2
Q2
2
1
Q2
A
Advanced Synoptic
A
B
M. D. Eastin
Dynamic Frontogenesis
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 gradient
enhances the cross-front pressure gradient
producing down-gradient cross-front flow
[enhances the cross-front circulation]
Advanced Synoptic
Coriolis torque turns the opposing down-gradient
cross-front flow into opposing along-front flow
[enhances the along-front vertical shear]
M. D. Eastin
Dynamic Frontogenesis
Conceptual Model: Example Case
1000-mb Isentropes
1000-mb Wind Barbs
1000-mb Frontogenesis
850-mb Q-vectors
Cold
Cold
Cold
Warm
Warm
Isentropes and Omega (ω)
N
S
850-mb Omega (ω)
850-mb Q-vectors
Advanced Synoptic
N
S
M. D. Eastin
Dynamic Frontogenesis
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 balance
and simultaneously enhance the thermal gradient
With no additional mechanism to offset the effects of ageostrophic
advection → rapid near-surface frontogenesis can occur!
Q2
Advanced Synoptic
Q2
M. D. Eastin
Q-vectors and Frontogenesis
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
Q-vectors
• 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
Advanced Synoptic
M. D. Eastin
Q-vectors and Frontogenesis
Example:
925-mb Q-vectors and Isentropes
11 November 2012 at 12 UTC
925-mb Isentropes
12 November 2012 at 00 UTC
Note: The regions of expected and observed frontogenesis / frontolysis generally agree
Part of the observed evolution is due to system motion and diabatic effects
Advanced Synoptic
M. D. Eastin
References
Bluestein, 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 Weather
Systems. 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 to
the 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 evolution
in midlatitude cyclones. Mon. Wea. Rev., 126, 1767-1791.
Advanced Synoptic
M. D. Eastin