Transcript Quasi-Geostrophic Theory: A Review of Basic Concepts

When advection destroys balance,
vertical circulations arise
ppt started from one by
James T. Moore
Saint Louis University
Cooperative Institute for
Precipitation Systems
Brian Mapes
COMET-MSC Winter Weather Course
29 Nov. - 10 Dec. 2004
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 dynamics
1. 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
Old school: Quasi-Geostrophic Omega Equation
(vorticity-oriented form)

 2 f 02  2 
f0 
1 2 
   
 
 Vg   g     Vg   

2   

 P 
 P
 
P  


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 z
(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).
Unsheared advection of T, u, v, vort, PV:
no problem, whole structure moves
cyclonic
(Trof)
Sheared advection
breaks thermal wind balance
cyclonic z
(Trof)
Sheared advection
breaks thermal wind balance
Z Trof
(hypsometric)
Sheared advection
breaks thermal wind balance
Z Trof
(hypsometric)
The PV view of balanced circulation:
(Rob Rogers’s fig)
Potential temperature and potential vorticity cross sections
Long-lived Great Plains MCV
Hurricane Andrew after landfall
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:



2
2


f

0
2
p 
2     2  Q, where
 P 



Vg

R  Vg


Q  Qx ,Qy  


T
,


T
p
p

P  x
y



or


 1

RP
Q  Qx ,Qy  
P0


 Vg

Vg


 x   p , y   p 


Q-vector Form of the Q-G Diagnostic Omega Equation

 2 f 02  2 
 p 
2     2  Q
 P 

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
Expanding Q and assuming adiabatic conditions yields the
following expression for Q:


Q  Qx ,Qy
Setting aside
the
coefficients,




Vg
RP 1  Vg

 
  p ,
  p 
 
y
P0  x

 u g  v g  
Qx   


 x x x y 
and
 u g  v g  
Qy   


 y x y y 
Interpretation of Qx
 u g  v g  
Qx   


 x x x y 
Geostrophic stretching
deformation weakens 
Geostrophic shearing
deformation turns 
cold
vg
ug
cold
warm
to
warm


cold
cold
warm

to+t

warm
Interpretation of Qy
 u g  v g  
Qy   


 y x y y 
Geostrophic shearing
deformation turns 
Geostrophic stretching
deformation strengthens 
cold
vg
ug
cold
warm
to
warm


cold
to+
t
cold
warm

warm

An Alternative form of Q in “natural” coordinates
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.
Defining the Orientation of Qs
and Qn with Respect to 
cold
Qn
Q
-1

+1
Qs

warm
+2
n
s
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.
Martin (1999, MWR)
Keyser
et al. (1992, MWR)
Defining Qn and Interpreting What It Means


   Vg

1    Vg

 
Qn 
    
    

   x  x

y  y

 


when
 0;
x


1   Vg


Qn 




  y  y

v g
1    u g
Qn 
i 

  y   y
y
1   v g  


Qn 
  y  y y 
  


j   
j  
i 
y  
  x
Defining Qn and Interpreting What It Means (cont.)
1   v g  


Qn 
  y  y y 

Qn

+1
+2
vg/y < 0; therefore Qn <0;
Couplets of div Qn:
• Tend to line up across the isotherms
• Show the ageostrophic response to the
geostrophically-forced packing/unpacking of
the isotherms
Qn points from cold to warm air; • Often exhibit narrow banded structures
typical of the “frontal” scale
confluence (diffluence) in wind
field implies frontogenesis
• Give an indication of how “active” a front
might be
(frontolysis)
Interpreting Q vectors: Qn
Advection by geostrophic stretching deformation acts to change
the magnitude of the thermal gradient vector, .
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)!
Low level wind: pure geostrophic
deformation (noting .Vg = 0), here
acting to weaken dT/dx.
cold
warm
Thermal wind
Upper level wind: add thermal wind to
low level wind. v component is positive
and decreases to north, so advection is
acting to increase upper-level v.
Defining Qs and Interpreting What It
 Means

  V

 V

1
Qs 



g
g

 
    
    

 y  x
 x  y
 

when
 0;
x


1   Vg


Qs 




  y  x

v g
1    u g
Qs 
i 

  y   x
x
1   v g  


Qs 
  y  x y 
  


j   
j  
i 
y  
  x
Defining Qs and Interpreting What It Means (cont.)
1   v g  


Qs 
  y  x y 
Thermal wind

Upper wind
+1

+2
Qss
Q
vg/x > 0; therefore Qs > 0.
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
Qs has cold air is to its left, causes
downstream of troughs
cyclonic rotation of the vector .
Thermal wind balance thus requires v • Are associated with the synoptic wave
to increase aloft, but geostrophic
scale of ascent and descent
advection acts to decrease v aloft.
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:


R T   Vg 
Q 
k 

P y 
x 
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:
A
B
-
B
A
Col Region
=
Q vectors
90 deg
Q
A
B
-
B
=
A
Jet Entrance
Region
90 deg
This is mainly the
cross-front, n
Q
Holton (1992)
component Qn
Q vectors in a setting where warm air rises
cold
Qn vectors
warm
Confluent Flow
Direct Thermal Circulation
Holton, 1992
Q vectors in a setting where COLD air rises
Jet Exit
Region
Q
Vageo
Thermally
Indirect
Circulation
Vageo
North
South
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.
Holton (1992)
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)
D
F
p
Dt
(S. Petterssen 1936)
 The 2-D scalar frontogenesis function (F ):
F > 0 frontogenesis, F < 0 frontolysis
 F: generalization of the quasi-geostrophic version, the Q-vector
Dg
F
p
Q
Dt etc.
Can also include diabatic heating gradients,
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 dynamics
1. 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: nomenclature
Schultz et al. MWR 1999
“The intricacies of instabilities”
Conditional Symmetric Instability: Cross section of es and Mg
taken normal to the 850-300 mb thickness contours
s
es-1
es
es+ 1
Mg +1
Mg
Horiz.
stable
Mg -1
Symm.
unstable
Vert.
stable
Note: isentropes of es
are sloped more vertical
than lines of absolute
geostropic momentum,
Mg.
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
Emanuel (1985, JAS)
Frontal secondary circ - with condensation on ascent
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
A Conceptual Model: Plan View of Key Processes
NW
SE
NW-SE cross-section shown on next slide.
Often found in the vicinity of an extratropical cyclone warm front, ahead of a longwave trough in a region of strong, moist, mid-tropospheric southwesterly flow
A Conceptual Model: Cross-Sectional View of Key
Processes
Dry Air
CSI
Convectively
Unstable
es
Arrows = Ascent zone
F = Frontogenesis zone
Heavy
Shaded area = CSI
snow area
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; e decreasing with
height)
– Conditional Symmetric Instability (CSI; lines
of es are more vertical than lines of constant
absolute geostrophic momentum or Mg)
– Weak Symmetric Stability (WSS; lines of es
are nearly parallel to lines of constant absolute
geostrophic momentum or Mg)
Spectrum of Mesoscale Instabilities
Nolan-Moore Conceptual Model
• 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.
Nolan-Moore Conceptual Model
• 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.
• 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 e 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 e or es are more vertical than the
Mg surfaces.
• In this way frontogenesis and the development of PSI/ CSI are linked.
Frontogenesis (definition)
D
F
p
Dt
(S. Petterssen 1936)
 The 2-D scalar frontogenesis function (F ):
F > 0 frontogenesis, F < 0 frontolysis
 F: generalization of the quasi-geostrophic version, the Q-vector
Dg
F
p
Q
Dt etc.
Can also include diabatic heating gradients,
Vector Frontogenesis Function
F  Fn n  Fs s
(Keyser et al. 1988, 1992)
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.
D
Fn  
 p
Dt
Change in direction (rotation)
D
 Corresponds to vertical motion on the scale of the
Fs  n  (k  p ) baroclinic wave itself: rotation of T gradient by a cyclone’s
winds causes along-front F vectors to converge on east side
Dt
of low pressure
Three-Dimensional Frontogenesis Equation
1
5
9
Terms 1, 5, 9: Diabatic Terms
2
4
3
6
7
8
10
11
12
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)
Assumptions to Simplify the Three-Dimensional Frontogenesis Equation
y’

x’
+ 1
+ 2
• 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
Simplified Form of the Frontogenesis Equation
d     u  v  w    d 
F 



 
 
dt  y   y  x  y  y  y  z y   dt 
A
B
Term A: Shear term
Term B: Confluence term
Term C: Tilting term
Term D: Diabatic Heating/Cooling term
C
D
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, 1991 Mid-Latitude Weather Systems
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
Carlson (Mid-latitude Weather Systems, 1991)
Frontogenesis: Tilting Term
Adiabatic cooling to north and warming to
south increases horizontal thermal gradient
Carlson, 1991 Mid-Latitude Weather Systems
Frontogenesis: Diabatic Heating/Cooling Term
frontogenesis
T constant
T increases
frontolysis
T increases
T constant
Carlson, 1991 Mid-Latitude Weather Systems
Frontogenesis/Frontolysis with Deformation with
No Diabatic Effects or Tilting Effects
d
1
F
     Def R cos 2  Div
dt
2
where:
  v u  2  u v  2 
Def R   
  
  
 x y  
  x y 
and
= angle between the isentropes
and the axis of dilatation
Petterssen (1968)
1
2
Kinematic Components of the Wind
Translation
Divergence
Vorticity
Deformation
Stretching and Shearing Deformation Patterns
Stretching
Shearing
Deformation
Deformation
Stretching Deformation Patterns
Stretching along
the flow
Translational component of
wind removed
Stretching normal
to the flow
Translational component of
wind removed
Bluestein (1992, Synoptic-Dynamic Met)
Shearing Deformation Patterns
Stretching in a
direction 45° to
the left of the flow
Translational component of
wind removed
Stretching in a
direction 45° to
the right of the flow
Translational component of
wind removed
Bluestein (1992, Synoptic-Dynamic Met)
 < 45°
Frontogenesis
Axis of dilatation
 > 45°
Frontolysis
Axis of dilatation
Petterssen (Weather Analysis
and Forecasting, vol. 1, 1956)
Pure Deformation Wind Field Acting on a Thermal Gradient
Isotherms are rotated and
brought closer together
Keyser et al. (MWR, 1988)
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.
Ageostrophic Adjustments in Response to
Frontogenetical Forcing
West
East
West
East
Frontogenetical Circulation
North
South
COLD
WARM
Thermally Direct Circulation
Carlson
1991)
Strength and
Depth(Mid-latitude
of the vertical Weather
circulationSystems,
is modulated
by static stability
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, MidLatitude 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
Defining Fs and Fn Vectors from the Frontogenesis Function
d 
F

dt

F  Fn n  Fs s
1 
Fn    div  Def R cos(2 )
2
1 
Fs    z  Def R sin(2 )
2
 


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.
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.
)
Augustine and Caracena (1994, WAF)