Kinematics of the horizontal wind field

1. Kinematics of the horizontal wind field (Kinematics: from the Greek word for ‘motion’, a description of the motion of a particular field without regard to how it came about or how it will evolve)

2. y N V v W E u x S To derive a mathematical expression for the key kinematic properties of the wind field we will use the coordinate system on the right. y We will use Taylor Expansion to estimate the wind field at an arbitrary point x,y from the wind at a nearby point x0, y0 x, y x0, y0

3. Peform a 2D Taylor expansion: For simplicity, lets assume that x0, y0 is the origin 0,0 And that we can obtain an adequate estimate of u,v by retaining only the first derivatives. We are assuming that over the small distance the u and v field vary linearly. Then…

4. Let’s take a simple step and write each derivative term as (for example) :

5. From before: (1) (2) Now we will write two nonsense equations (3) (4) Now we add (1) and (3). We also separately add (2) and (4). Then we rearrange the terms and get…………

6. Translation Divergence Shearing Deformation Relative Vorticity Stretching Deformation Any wind field that varies linearly can be characterized by these five distinct properties. Non-linear wind fields can be closely characterized by these properties.

7. y x Translation The effect of translation on a fluid element: Change in location, no change in area, orientation, shape

8. y Divergence (d > 0) Convergence (d < 0) The effect of convergence on a fluid element: x Change in area, no change in orientation, shape, location

9. y Positive (cyclonic) vorticity ( > 0). Negative (anticyclonic) vorticity ( < 0) The effect of negative vorticity on a fluid element: x Change in orientation, no change in area, shape, location

10. y E-W Stretching Deformation (D1 > 0). N-S Stretching Deformation (D1 < 0). The effect of stretching deformation on a fluid element: x Change in shape, no change in area, orientation, location

11. y SW-NE Shearing Deformation (D1 > 0). NW-SE Shearing Deformation (D1 < 0). The effect of shearing deformation on a fluid element: x Change in shape, no change in area, orientation, location

12. Why are we interested in these properties? Net Divergence in an air column leads to the development of low surface pressure Net Convergence in an air column leads to the development of high surface pressure L H

13. Vertical vorticity (spin about a vertical axis) arises from three sources: Horizontally sheared flow, flow curvature, and the rotation of the earth. Relative vorticity: shear and curvature. Absolute vorticity: shear, curvature and earth rotation. z < 0 z > 0 z < 0 z > 0

14. Absolute vorticity allows us to identify short waves and shear zones within the jetstream. Short waves trigger cyclogenesis and can help trigger deep convection in the warm season.

15. Positive Vorticity Advection on a 500 mb map can be used as a proxy for divergence aloft, and is related to the development of low surface pressure and upward air motion.

16. 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 Deformation flow is fundamental to the development of fronts Time = t + Dt Time = t y y x x

17. EXAMPLES OF DEFORMATION Axis of Dilitation

18. EXAMPLES OF DEFORMATION Axis of Dilitation

19. CONFLUENT and DIFLUENT FLOW Is this flow convergent? Is this flow divergent? NO: The areas of the two boxes are identical. The flow is a combination of translation and deformation.

20. The terms for divergence, relative vorticity, and deformation strictly apply on a plane tangent to the earth’s surface. If we take earth’s curvature into account, we have to add an additional term.

21. Suppose the wind is southerly and uniform. Is the wind convergent? Red = wind Blue = wind component y Yes! y y x x x Convergence of meridians toward north leads to convergence. This is the earth curvature term (the last term) in the expression for convergence (d).

22. Suppose the wind is westerly and uniform. Does vorticity exist? Yes! Convergence of meridians toward north creates vorticity. This is the earth curvature term (the last term) in the expression for vorticity ().

23. In a similar way, convergence of the earth’s meridians toward the north leads to deformation in otherwise uniform flow Earth’s curvature terms are an order of magnitude smaller than other terms, but cannot be ignored in models, at least in the middle and high latitudes.