Section 8 Vertical Circulation at Fronts

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. 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 and Wehave Substitute hydrostatic equation and equate expressions: Solve for the slope of the front

6. 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 diabatic heating or cooling rate Temperature gradient 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 Shearing deformation ) Stretching Deformation Deformation acting on temperature gradient Deformation acting on temperature gradient

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

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

22. T- 8DT T- 8DT T- 7DT T- 7DT T- 6DT T- 6DT T- 5DT T- 5DT T- 4DT T- 4DT T- 3DT T- 3DT T- 2DT T- 2DT T- DT T- DT T T Shearing Deformation y y x x

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 • Of vertical • Gradient (N-S direction) Weighting factor Magnitude of q gradient in one direction Magnitude of total q gradient

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

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 motion Before After z z q+4Dq q+4Dq q+2Dq q q+2Dq q x or y x or y

31. 2D Frontogenetic Function ( )

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

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. y y x x This expression can be reduced to: 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. T- 8DT T- 8DT T- 7DT T- 7DT T- 6DT T- 6DT T- 5DT T- 5DT T- 4DT T- 4DT T- 3DT T- 3DT T- 2DT T- 2DT T- DT T- DT T T • 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 x x

39. 3. S.G. vs. Q.G. Approximations • Q.G.: S.G. where

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 warm warm cold

42. 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 Warm air Cold air (heating in the warm side and cooling in the cold side) will produce A thermally direct circulation and promote Frontogenesis. Isentrope Warm air Cold air

43. Geostrophic shearing deformation Geostrophic stretching deformation

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 frontogenesis and a direct circulation in which warm air is rising and cold air sinking. Geostrophic shearing deformation confluent flow along front

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 stable neutral neutral unstable unstable