Dynamics of Rotation, Propagation, and Splitting

1. Dynamics of Rotation, Propagation, and Splitting METR 4433: Mesoscale MeteorologySpring 2013 SemesterAdapted from Materials by Drs. Kelvin Droegemeier, Frank Gallagher III and Ming Xue; and from Markowski and Richardson (2010)School of MeteorologyUniversity of Oklahoma

2. Dynamics of Isolated Updrafts • Linear theory is a powerful tool for understanding storm dynamics! • It can be used to explain • Origin of mid-level rotation • Mesocyclone intensification • Deviate motion and propagation • Nonlinear theory is needed to explain • Splitting • We’ve looked at these qualitatively and now will apply theory

3. Origin of Mid-Level Rotation • We already established that mid-level rotation is a result of the titling by an updraft of horizontal vorticity associated with environmental shear • We’re now going to look at theory, which leads us into the concept of streamwise vorticity

4. Origin of Mid-Level Rotation • Begin with vertical vorticity equation

5. Origin of Mid-Level Rotation

6. Origin of Mid-Level Rotation • Now move into a storm-relative reference frame, where C is the storm motion and V-C the storm-relative wind

7. Origin of Mid-Level Rotation Horizontal Vorticity

8. Origin of Mid-Level Rotation • Tilting generates vertical vorticity, with the vortex coupled straddling the updraft • Once vertical vorticity is present, it can then be advected – with the only wind that matters  the STORM-RELATIVE wind!!

9. Origin of Mid-Level Rotation Horizontal Vorticity

10. Example of Crosswise Vorticity

11. Example of Crosswise Vorticity

12. Example of Crosswise Vorticity • Note two updrafts: The “hill,” which is the primary updraft, and the vertical motion induced by storm-relative flow in conjunction with it

13. Example of Crosswise Vorticity • The net updraft (black) and vertical vorticity (red). The storm-relative winds are zero at this early stage because the updraft is moving along the hodograph (red dot)

14. Example of Streamwise Vorticity

15. Example of Streamwise Vorticity

16. Example of StreamwiseVorticity • Note two updrafts: The “hill,” which is the primary updraft, and the vertical motion induced by storm-relative flow in conjunction with it

17. Example of Crosswise Vorticity • The net updraft (black) and vertical vorticity (red). The storm-relative winds at low-levels are from the south: draw line from red dot (storm motion) back to the hodograph

18. Idealized Hodograph • Note locations of streamwise and crosswise vorticity depending upon storm-relative winds Storm Motion

19. Idealized Hodograph • Note locations of streamwise and crosswise vorticity depending upon storm-relative winds Storm-Relative Winds Storm Motion

20. Idealized Hodograph • Note locations of streamwise and crosswise vorticity depending upon storm-relative winds Storm-Relative Winds Storm Motion

21. Streamwise Vorticity • It is the vorticity in the direction of the unit vector storm-relative wind • The numerator is called the Helicity Density, as noted previously in class

22. Relative Helicity • The Relative Helicity, or Normalized Helicity Density, is just the streamwisevorticity normalized by the magnitude of the vorticity, or • Note that • Where theta is the angle between the vorticity and storm-relative velocity vectors

23. Relative Helicity • Dividing by the magnitude of the vorticity vector yields the relative helicity • It’s clear that Relative Helicity is simply the cosine of the angle between the vorticity and storm-relative velocity vectors and varies between -1 and +1

24. Optimal Conditions for a Mesocyclone • Optimal conditions for a mesocyclone are • Streamwisevorticity (alignment between storm-relative winds and environmental horizontal vorticity) – that is, Relative Helicity close to 1 • Strong storm-relative winds • BOTH conditions must be met • Can quantify these two effects theoretically

25. Optimal Conditions for a Mesocyclone • r = correlation coefficient between w and vertical vorticity • P is proportional to updraft growth rate

26. Optimal Conditions for a Mesocyclone • The cosine term is called the relative helicity (cosine of angle between the storm-relative wind vector and the horizontal vorticity vector). It is the fraction of horizontal vorticity that is streamwise. When cosine term is zero, horizontal inflow vorticity is purely crosswise.

27. Optimal Conditions for a Mesocyclone • Note that alignment of the horizontal vorticity vector and storm-relative wind vector is NOT SUFFICIENT. One must have strong storm-relative winds to co-locate updraft and vertical vorticity(via the P term).

28. Testing the Theory with a 3D Cloud Model Droegemeier et al. (1993)

29. Testing the Theory with a 3D Cloud Model Droegemeier et al. (1993)

30. Testing the Theory with a 3D Cloud Model Actual Theoretical Droegemeier et al. (1993)

31. Testing the Theory with a 3D Cloud Model Actual Droegemeier et al. (1993)

32. Testing the Theory with a 3D Cloud Model Actual Droegemeier et al. (1993)

33. Testing the Theory with a 3D Cloud Model Notice how the correlation betweenvertical velocity and vertical vorticityincreases over time as the vorticityand velocity contours begin tooverlap. Droegemeier et al. (1993)

34. Note the Large Relative Helicity Isn’t Enough – Need Storm Storm-Relative Winds as Well Relative Helicity Droegemeier et al. (1993)

35. Testing the Theory with a 3D Cloud Model The rule of thumb of 90 degrees ofturning and at least 10 m/s of storm-relative winds in the 0-3 kmlayer holds true Droegemeier et al. (1993)

36. Updraft Splitting • We discussed previously updraft splitting and the role of precipitation, noting that storms split in 3D cloud models even when precipitation is “turned off” • Now we look at the dynamics of splitting

37. Dynamics of Isolated Updrafts We want to obtain an expression for p’ = Stuff.... Note that we did this before (spin and splat)!

38. Dynamics of Isolated Updrafts

39. Dynamics of Isolated Updrafts

40. Dynamics of Isolated Updrafts Can you spot the nonlinear versus linear terms?

41. Dynamics of Isolated Updrafts

42. Nonlinear Theory of an Isolated Updraft

43. Note that low pressure exists at the center of each vortex and thus “lifting pressure gradients” cause air to risefrom high to low pressure, enhancing the updraft beyondbuoyancy effects alone and leading to splitting

44. Note that low pressure exists at the center of each vortex and thus “lifting pressure gradients” cause air to risefrom high to low pressure, enhancing the updraft beyondbuoyancy effects alone and leading to splitting

45. Dynamics of Isolated Updrafts

46. Nonlinear Theory of an Isolated Updraft

47. Nonlinear Theory of an Isolated Updraft

48. Selective Enhancement and Deviate Motion of Right-Moving Storm • For a purely straight hodograph (unidirectional shear, e.g., westerly winds increasing in speed with height and no north-south wind present), an incipient supercell will form mirror image left- and right-moving members

49. Straight Hodograph: Idealized

50. Selective Enhancement and Deviate Motion of Right-Moving Storm • For a curved hodograph, the southern member of the split pair tends to be the strongest • It also tends to slow down and travel to the right of the mean wind