1 Dynamics of Rotation, Propagation, and Splitting METR 4433: Mesoscale Meteorology Spring 2013 Semester Adapted from Materials by Drs. Kelvin Droegemeier, Frank Gallagher III and Ming Xue; and from Markowski and Richardson (2010) School of Meteorology University 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 Now move into a storm-relative reference frame, where C is the storm motion and V-C the storm-relative wind Origin of Mid-Level Rotation

7 Horizontal Vorticity

8 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!! Origin of Mid-Level Rotation

9 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 Streamwise Vorticity 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 Motion Storm-Relative Winds

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

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

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

23 Dividing by the magnitude of the vorticity vector yields the 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 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 Relative Helicity

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

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

26 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. 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. Optimal Conditions for a Mesocyclone

27 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). 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). Optimal Conditions for a Mesocyclone

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 Droegemeier et al. (1993) Theoretical Actual

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

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

33 Testing the Theory with a 3D Cloud Model Droegemeier et al. (1993) Notice how the correlation between vertical velocity and vertical vorticity increases over time as the vorticity and velocity contours begin to overlap.

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

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

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....

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 rise from high to low pressure, enhancing the updraft beyond buoyancy 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 rise from high to low pressure, enhancing the updraft beyond buoyancy 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

51 Curved Hodograph – Selective Enhancement of Cyclonic Updraft

52 Obstacle Flow – Wrong! Newton and Fankhauser (1964)

53 Magnus Effect – Wrong! Newton and Fankhauser (1964) Storm Updraft Fast Slow H L Via Bernoulli effect, low pressure located where flow speed is the highest, inducing a pressure gradient force that acts laterally across the updraft

54 Dynamics of Isolated Updrafts

55 Linear Theory of an Isolated Updraft

56 Linear Theory of an Isolated Updraft  This equation determines where pressure will be high and low based upon the interaction of the updraft with the environmental vertical wind shear Rotunno and Klemp (1982)

57 Vertical Wind Shear East Up Shear = V(upper) – V(lower)

58 Storm Updraft (w>0) Linear Theory of an Isolated Updraft y x P’>0 P’<0 Rotunno and Klemp (1982)

59 Storm Updraft (w>0) Unidirectional Shear (Straight Hodograph)  Note that if the shear vector is constant with height (straight hodograph), the high and low pressure centers are identical at all levels apart from the intensity of w Storm Updraft (w>0) Storm Updraft (w>0) Low Mid Upper Low MidUpper P’>0 P’<0 P’>0 P’<0 P’>0 P’<0

60 Straight Hodograph S S Rotunno and Klemp (1982)

61 Straight Hodograph: Idealized

62 Straight Hodograph: Real

63 Storm Updraft (w>0) Turning Shear Vector P’>0 P’<0  Note that if the shear vector turns with height (curved hodograph), so do the high and low pressure centers Storm Updraft (w>0) Storm Updraft (w>0) Low Mid Upper Low MidUpper P’<0 P’>0 P’<0

64 Curved Hodograph S S S S Rotunno and Klemp (1982)

65 Curved Hodograph – Selective Enhancement of Cyclonic Updraft

66 Curved Hodograph – Selective Enhancement of Cyclonic Updraft

67 Predicting Thunderstorm Type: The Bulk Richardson Number n Need sufficiently large CAPE (2000 J/kg) n Denominator is really the storm-relative inflow kinetic energy (sometimes called the BRN Shear) n BRN is thus a measure of the updraft potential versus the inflow potential

68 Results from Observations and Models

69 General Guidelines for Use

70 BRN in the Modeling Study Droegemeier et al. (1993)

71 BRN in the Modeling Study Droegemeier et al. (1993)

72 BRN in the Modeling Study Droegemeier et al. (1993)

73 Supercell Longevity/Predictability  Observations show that supercell storms are relatively long-lived and thus more easily predictable than their weaker-shear, weakly-rotating counterparts 33 min Forecast Low-level Reflectivity Observed Low-level Reflectivity

74 It has been proposed that storms having high helicity (rotating updrafts) are resilient to turbulent decay and thus live longer It has been proposed that storms having high helicity (rotating updrafts) are resilient to turbulent decay and thus live longer Consider the 3D vector vorticity equation derived earlier Consider the 3D vector vorticity equation derived earlier Consider also the idealized situation in which the vector velocity is exactly parallel to the vector vorticity and differs only by a constant (called the abnormality, or lambda) Consider also the idealized situation in which the vector velocity is exactly parallel to the vector vorticity and differs only by a constant (called the abnormality, or lambda) Helicity

75 Such a flow, in the absence of baroclinic effects and friction, is called a Beltrami flow – and is purely helical. Under these conditions, it is easy to show that Such a flow, in the absence of baroclinic effects and friction, is called a Beltrami flow – and is purely helical. Under these conditions, it is easy to show that Helicity

76 In a Beltrami flow (and valid for supercells, with caveats), the nonlinear advection exactly cancels stretching plus tilting In a Beltrami flow (and valid for supercells, with caveats), the nonlinear advection exactly cancels stretching plus tilting Because advection and stretching create small scales (cascade), the downscale cascade of energy is effectively blocked, possibly leading to longer-lived storms Because advection and stretching create small scales (cascade), the downscale cascade of energy is effectively blocked, possibly leading to longer-lived storms Helicity Nonlinear Vorticity Advection StretchingTilting

77 In fluid dynamics, helicity typically is integrated over a volume. In storm dynamics, the velocity of significance is the storm-relative wind, and thus helicity is not Galilean invariant (depends upon storm motion) In fluid dynamics, helicity typically is integrated over a volume. In storm dynamics, the velocity of significance is the storm-relative wind, and thus helicity is not Galilean invariant (depends upon storm motion) We also are concerned about storm inflow, so helicity is typically computed over the inflow layer (0-3 km) and is termed Storm-Relative Environmental Helicity (SREH) We also are concerned about storm inflow, so helicity is typically computed over the inflow layer (0-3 km) and is termed Storm-Relative Environmental Helicity (SREH) Helicity

78 Storm Relative Environmental Helicity  SREH is the area swept out by the S-R winds between the surface and 3 km  It includes all of the key ingredients mentioned earlier  It is graphically easy to determine

79 Storm Relative Helicity Storm Motion SFC 1 km 2 km 3 km 4 km 5 km 6 km 7 km This area represents the 1-3 km helicity SREHPotential Tornado Strength m 2 s -2 Weak m 2 s -2 Strong > 450 m 2 s -2 Violent

80 Computing helicity from wind data is easy, once storm motion is either known or assumed based upon environmental winds (see also Eq in the text) Computing helicity from wind data is easy, once storm motion is either known or assumed based upon environmental winds (see also Eq in the text) Helicity

81 SREH in the Modeling Study Droegemeier et al. (1993)

82 SREH in the Modeling Study Droegemeier et al. (1993)

83 SREH in the Modeling Study Droegemeier et al. (1993)

84 SREH in the Modeling Study Droegemeier et al. (1993)

85 SREH in the Modeling Study Droegemeier et al. (1993)

86 Real Data

87 Streamwise vorticity is a key ingredient in supercell dynamics; however, the alignment between the vertical velocity and vertical vorticity vectors is insufficient – also need strong storm-relative winds and turning of the wind shear vector with height Updraft splitting is principally the result of nonlinear dynamics in the form of lifting pressure gradients on the flanks of the storm Deviate updraft motion is principally the result of linear dynamics in the form of lateral pressure gradient forces associated with the turning of the environmental shear vector with height Helicity is believed responsible for the longevity of supercells The Bulk Richardson Number is a good predictor of storm type Storm-Relative Environmental Helicity is a good predictor of storm type and net updraft rotation, including sign Key Summary Points