1. Section 8 Vertical Circulation at Fronts
Structural and dynamical characteristics of mid-latitude fronts
Frontogenesis
Semi-geostrophic equations
Symmetric instability

2. 1. Structure and dynamical characteristics of mid-latitude fronts

3. Larger than background relative vorticity
EXAMPLES OF FRONTS
A front is a transition zone between different air masses. It is characterized by:
Larger than background horizontal temperature (density) contrasts ( strong vertical shear)
Larger than background relative vorticity
Larger than background static stability
a quasi linear structure (length >> width)

4. Let’s for the moment consider a zero-order front
We will assume that: 1) front is parallel to x axis
2) front is steady-state
3) pressure is continuous across the front
4) density and T are discontinuous across the front

5. Warm side of front Cold side of front We have and
Substitute hydrostatic equation and equate expressions:
Solve for the slope of the front

6. Substituting geostrophic wind relationship
For cold air to underlie warm air, slope must be positive
1) Across front pressure gradient on the cold side must be larger that the pressure gradient on the warm side
Substituting geostrophic wind relationship
2) Front must be characterized by positive geostrophic relative vorticity
The stronger the density (T) contrast becomes, the stronger is the vorticity at the front.

7. First-order fronts
Larger than background horizontal temperature (density) gradient
Larger than background relative vorticity
3) Larger than background static stability

8. Working definition of a cold or warm front
The leading edge of a transitional zone that separates advancing cold (warm) air from warm (cold) air, the length of which is significantly greater than its width. The zone is characterized by high static stability as well as larger-than-background temperature gradient and relative vorticity.

9. 2. Frontogenetic Function
the Lagrangian rate of change of the magnitude of potential temperature gradient

10. Move to the whiteboard and talk about 1D frontogenesis

11. 3D Frontogenesis Expanding the total derivative
expanding the term involving the magnitude of the gradient

12. ( ) The solution The Three-Dimensional Frontogenesis Function becomes
Compared to

13. Confluence terms (or stretching deformation): with
Shearing terms (or shearing deformation):
involved with
Tilting terms: with derivative of omega

14. ( ) The terms in the yellow box all contain the derivative
which is the diabatic heating rate. These terms are
called the diabatic terms.

15. ( ) Horizontal gradient in Temperature gradient
diabatic heating or cooling rate
If and have the same sign, it means the diabatic heating will increase the temperature gradient.

17. q+3Dq
q+2Dq
q+Dq q
q+3Dq q
Vertical cross section of potential temperature

18. ( ) The terms in this yellow box represent the contribution
to frontogenesis due to horizontal deformation flow.

19. Stretching Deformation(
Stretching
deformation
Shearing
deformation )
Stretching Deformation
Deformation
acting on
temperature gradient
Deformation
acting on
temperature gradient

20. Stretching Deformation
Time = t y
Time = t + Dt T
T- DT
T- 2DT
T- 3DT
T- 4DT
T- 5DT
T- 6DT
T- 7DT
T- 8DT y
T- 8DT
T- 7DT
T- 6DT
T- 5DT
T- 4DT x x
T- 3DT
T- 2DT
T- DT T

21. ( ) Shearing Deformation Deformation acting on temperature gradient
Stretching
deformation
Shearing
deformation )
Shearing Deformation
Deformation
acting on
temperature gradient
Deformation
acting on
temperature gradient

22. Shearing Deformation x y y x T- 8DT T- 7DT T T- DT T- 2DT T- 3DT

23. ( ) The terms in this yellow box represent the contribution
to frontogenesis due to tilting.

24. ( ) Tilting terms Tilting Of vertical Gradient (E-W direction) Tilting
(N-S direction)
Weighting factor
Magnitude of q gradient in one direction
Magnitude of total q gradient

25. Tilting terms After x or y z Before z x or y q+2Dq q q+4Dq q+4Dq q+2Dq

26. ( ) The terms in this yellow box represent the contribution
to frontogenesis due to vertical shear acting on a
horizontal temperature gradient.

27. ( ) Vertical shear acting on a horizontal temperature gradient
(also called vertical deformation term)
Vertical shear of E-W wind
Component acting on
a horizontal temp gradient in x
direction
Vertical shear of N-S wind
component acting on
a horizontal temp gradient in y
direction

28. Vertical shear acting on a horizontal temperature gradient
Before
After q
q+3Dq
q+6Dq
q+9Dq
q+6Dq
q+9Dq
q+3Dq q z z x x

29. ( ) The term in this yellow box represents the contribution
to frontogenesis due to divergence.
Compression
of vertical
Gradient
by differential
vertical motion

30. Differential vertical motionq
After
x or y z
Before z
q+4Dq
q+4Dq
q+2Dq
q+2Dq q
x or y

31. 2D Frontogenetic Function( )

32. The stretching and shearing deformations “look like” one another:y y
T- 8DT
T- 7DT
T- 6DT
T- 8DT
T- 5DT
T- 7DT
T- 6DT
T- 4DT x
T- 5DT
T- 3DT
T- 4DT x
T- 3DT
T- 2DT
T- 2DT
T- DT
T- DT T T

33. Another view of the 2D frontogenesis function
Recall the kinematic quantities: divergence (D)
vorticity ()
stretching deformation (F1)
shearing deformation (F1).
and note that:
Substituting:

34. Shearing and stretching deformation
This expression can be reduced to: x y x y
Shearing and stretching deformation
“look alike” with axes rotated

35. We can simplify the 2D frontogenesis equation by rotating our coordinate axes to align with the axis of dilatation of the flow (x´)
where F is the total deformation

36. This equation illustrates that horizontal frontogenesis is only associated with divergence and deformation, but not vorticity

37. Note that
Where F is the total deformation of the flow, β is the angle between the isentropes and the dilatation axis of the total deformation field, and D is divergence (D <0 for convergence)

38. if non-zero is coincident with convergence (D<0)
Frontogenesis occurs
if non-zero is coincident with convergence (D<0)
if the total deformation field (F) acts on isentropes that are between 0 and 45° of the dilatation axis of the total deformation. deformation. y y
T- 8DT
T- 8DT
T- 7DT
T- 7DT
T- 6DT
T- 6DT
T- 5DT
T- 5DT
T- 4DT x
T- 4DT x
T- 3DT
T- 3DT
T- 2DT
T- 2DT
T- DT
T- DT T T

39. 3. S.G. vs. Q.G. Approximations
Q.G.: S.G.
where
S.G.: assumes geostrophic flow in the along-the-front direction and allows ageostrophic flow in the across-front direction.

40. Sawyer-Eliassen Equation
Geostrophic deformation
Diabatic heating
Right side of equation represent the forcing
(known from measurements or in model solution)
, the streamfunction, is the response.
V and ω can be derived from 
Questions: 1) How is the thermal wind balance maintained by the transverse circ.?
2) Where should we expect upward motion (precipitation)?

41. Cold
Frontogenesis  larger temperature gradient  stronger vertical wind shear (Ug)  southward Va decelerates Ug in the low levels and northward Va accelerates Ug in the upper levels.
Left: . Upper level (500 hPa) weather map of 6 August 1996, 00 UTC. The temperature (°C) and the wind vector as measured by radiosonde are indicated. The contours represent isopleths of the 500- hPa height (labeled in dm; contour interval is 2.5 dm). (Cold air is located in the northeast while warm air is located in the south west, i.e. isotherms are oriented approximately parallel to the axis of dilatation)
Right: The secondary (vertical) circulation forced by the deformation field in a frontal region. The dash-dotted lines represent isentropes in the plane z=0. Also shown in this plane is the deformation field (the arrows). U is the geostrophic flow (see the text).
warm
warm
cold

42. (heating in the warm side and cooling in the cold side) will produce
Nature of the solution of the Sawyer-Eliassen Equation:
A direct circulation (warm air rising and cold air sinking) will result with positive forcing.
An indirect circulation (warm air sinking and cold air rising) will result with negative forcing.
Isentrope
Cold air
Warm air
(heating in the warm side
and cooling in the cold side) will produce
A thermally direct circulation and promote
Frontogenesis.
Isentrope
Cold air
Warm air

43. Geostrophic
shearing
deformation
stretching

44. Geostrophic stretching deformation
Entrance region of jet
Note in this figure that both and are negative, implying
frontogenesis and a direct circulation in which warm air is rising and
cold air sinking.

45. Note in this figure that both and are positive, implying
Geostrophic shearing deformation
confluent flow
along front
Note in this figure that both and are positive, implying
frontogenesis and a direct circulation in which warm air is rising and
cold air sinking.

46. The spatial variation of the coefficients and the presence of the cross- derivative term produce a distortion of the secondary circulation. The frontal zone slopes toward the cold air side with height; there is an intensification of the cross-frontal flow near the surface in the region of large absolute vorticity on the warm air side of the front, and a tilting of the circulation with height.

48. Why does a spinning top stay upright?

49. Buoyancy tends to stabilize air parcels against vertical displacements, and rotation tends to stabilize parcels with respect to horizontal displacements.
If ordinary static and inertial stabilities are satisfied, is the flow always stable?

50. 4. Symmetric Instability
hydrostatic instability
Inertial instability
stable
neutral
unstable
stable
neutral
unstable
Instability with respect to vertical displacements is referred to as hydrostatic (or simply, static) instability. Instability with respect to horizontal displacements, however, is referred to as inertial instability

51. MSI: an intuitive explanation
M = absolute zonal momentum 30 40
M = fy-ug
dM/dy>0 60 70
see also: Jim Moore’s meted module on frontogenetic circulations & stability)

52. - Dash: qe - Solid: Mg - - - - - - - - Potential
Potential Symmetric Stability
Potential Symmetric INstability -
Dash: qe
Solid: Mg - - - - - - - - -

53. Symmetric instability evaluation
The flow satisfies and z z z
Stable
Neutral
Unstable y y y