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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. COMETMSC Winter Weather Course 29 Nov.  10 Dec. 2004. QuasiGeostrophic Theory.
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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
COMETMSC Winter Weather Course
29 Nov.  10 Dec. 2004
Old school: QuasiGeostrophic Omega Equation
(vorticityoriented form)
A B C
Term A: threedimensional 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
PV view of how maintenance of balance requires vertical motions
warm
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 z
(Trof)
COOL
CORE
Unshearedadvection of T, u, v, vort, PV:
no problem, whole structure moves
warm
cyclonic
(Trof)
COOL
CORE
warm
cyclonic z
(Trof)
COOL
CORE
Coriolis forces
COOL
CORE
Z Trof
(hypsometric)
imbalanced force = acceleration
COOL
CORE
Z Trof
(hypsometric)
The PV view of balanced circulation:
(Rob Rogers’s fig)
Potential temperature and potential vorticity cross sections
Longlived Great Plains MCV
Hurricane Andrew after landfall
Qvector Form of the QG Diagnostic Omega Equation
Alternate approach developed by Hoskins et al. (1978, Q. J.) – manipulated the equations so forcing is 1 term, not 2:
Qvector Form of the QG Diagnostic Omega Equation
Treat Laplacian as a “sign flip” Then,
If 2•Q > 0 (convergence of Q)then w < 0 (upward vertical motion)
If 2•Q < 0 (divergence of Q)then w > 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
Interpreting Q Vectors
Expanding Q and assuming adiabatic conditions yields the following expression for Q:
Setting aside the coefficients,
Interpretation of Qx
Geostrophic stretching deformation weakens
Geostrophic shearing deformation turns
cold
vg
ug
cold
warm
to
warm
cold
to+t
cold
warm
warm
Interpretation of Qy
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
Qn
Q
cold
1
Qs
+1
+2
n
warm
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)
DefiningQn and Interpreting What It Means
DefiningQn and Interpreting What It Means (cont.)
+1
+2
Qn
vg/y < 0; therefore Qn <0;
Qn points from cold to warm air; confluence (diffluence) in wind field implies frontogenesis (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
Upper level wind: addthermal windtolow levelwind. v component is positive and decreases to north, so advection is acting to increase upperlevel v.
Thermal wind
DefiningQs and Interpreting What It Means
Thermal wind
Upper wind
DefiningQs and Interpreting What It Means (cont.)
+1
+2
Qs
Qs
vg/x > 0; therefore Qs > 0.
Qs has cold air is to its left, causes cyclonic rotation of the vector . Thermal wind balance thus requires v to increase aloft, but geostrophic 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:
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

Col Region
B
A
=
Q vectors
90 deg
Q
A
B
B
A

Jet Entrance Region
=
90 deg
This is mainly the crossfront, n component Qn
Q
Holton (1992)
Q vectors in a setting where warm air rises
cold
Qn vectors
warm
Direct Thermal Circulation
Confluent Flow
Holton, 1992
Q vectors in a setting where COLD air rises
Jet Exit Region
Q
Vageo
Thermally Indirect Circulation
Vageo
South
North
Idealized pattern of sealevel isobars (solid) and isotherms (dashed) for a train of cyclones and anticyclones. Heavy bold arrows are Q vectors. This is mostly the alongfront or s component Qs.
Holton (1992)
is met (translation: PV must be positive, so that the system is symmetrically stable)
(S. Petterssen 1936)
g
Q
Symmetric instabilities, contributing to banded precipitation, often north and east of midlatitude cyclones
Conditional Symmetric Instability: Cross section of esand Mg taken normal to the 850300 mb thickness contours
s
es1
es
es+ 1
Mg +1
Symm.
unstable
Note: isentropes of es
are sloped more vertical
than lines of absolute
geostropic momentum,
Mg.
Mg
Vert.
stable
Horiz.
stable
Mg 1
Conditional Symmetric Instability in the Presence of Synoptic Scale Lift – Slantwise Ascent and Descent
Multiple Bands with Slantwise Ascent
Frontal secondary circulation  constant stability
Emanuel (1985, JAS)
Frontal secondary circ  with condensation on ascent
Schematic of ConvectiveSymmetric Instability Circulation
Blanchard, Cotton, and Brown, 1998 (MWR)
ConvectiveSymmetric Instability
Multiple Erect Towers with Slantwise Descent
Sanders and Bosart, 1985: Mesoscale Structure in the Megalopolitan Snowstorm of 1112 February 1983. J. Atmos. Sci.,42, 10501061.
A Conceptual Model: Plan View of Key Processes
NW
SE
NWSE crosssection 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, midtropospheric southwesterly flow
A Conceptual Model: CrossSectional View of Key Processes
Dry Air
CSI
Convectively Unstable
es
Heavy snow area
Arrows = Ascent zone
F = Frontogenesis zone
Shaded area = CSI
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
Spectrum of Mesoscale Instabilities
Figure from Nicosia and Grumm(1999,WAF).Potential symmetric instability occurs where the midlevel dry tongue jet overlays the lowlevel easterly jet (or cold conveyor belt), north of the surface low. In this area dry air at midlevels overruns moistureladen lowlevel easterly flow, thereby steepening the slope of the e surfaces.
(S. Petterssen 1936)
g
Q
(Keyser et al. 1988, 1992)
ThreeDimensional Frontogenesis Equation
1
2
4
3
5
6
7
8
9
10
11
12
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 (SynopticDynamic Met. In MidLatitudes, vol. II, 1993)
Assumptions to Simplify the ThreeDimensional Frontogenesis Equation
y’
+ 1
x’
+ 2
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 MidLatitude Weather Systems
Frontogenesis: Confluence Term
Cold advection to the north
Warm advection to the south
Carlson, 1991 MidLatitude 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 (Midlatitude Weather Systems, 1991)
Frontogenesis: Tilting Term
Adiabatic cooling to north and warming to south increases horizontal thermal gradient
Carlson, 1991 MidLatitude Weather Systems
Frontogenesis: Diabatic Heating/CoolingTerm
frontogenesis
T constant
T increases
frontolysis
T increases
T constant
Carlson, 1991 MidLatitude Weather Systems
Frontogenesis/Frontolysis with Deformation with No Diabatic Effects or Tilting Effects
where:
and
= angle between the isentropes and the axis of dilatation
Petterssen (1968)
Kinematic Components of the Wind
Vorticity
Translation
Divergence
Deformation
Stretching and Shearing Deformation Patterns
Stretching
Deformation
Shearing
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, SynopticDynamic 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, SynopticDynamic 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)
Carlson (MidLatitude Weather Systems, 1991)
Ageostrophic Adjustments in Response to Frontogenetical Forcing
West
East
West
East
Frontogenetical Circulation
North
South
COLD
WARM
Thermally Direct Circulation
Carlson (Midlatitude Weather Systems, 1991)
Strength and Depth of the vertical circulation is modulated by static stability
Carlson (Midlatitude Weather Systems,1991)
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
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
Greg Mann, 2004
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)
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 lowlevel jet.
)
Augustine and Caracena (1994, WAF)