Can a gust front tilt horizontal vortex lines to produce a tornado?

1. Can a gust front tilt horizontal vortex lines to produce a tornado? Paul Markowski Pennsylvania State University Bob Davies-Jones Emeritus, NSSL

2. Helmholtz’ (1858) theorem • Consider two parcels following the same trajectory. Also assume that the parcels lie on the same vortex line. In an inviscid, barotropic flow, vortex lines move as material lines (vortex lines are ‘‘frozen’’ into the fluid and behave like elastic strings that the flow moves, stretches, and reorients) Hermann von Helmholtz (1821–1894) • At some later time, the parcels have moved and the segment of the vortex line moves as a material line according to the trajectories of the parcels. A B

3. Helmholtz’ (1858) theorem In an inviscid, barotropic flow, vortex lines move as material lines (vortex lines are ‘‘frozen’’ into the fluid and behave like elastic strings that the flow moves, stretches, and reorients) • Consider two parcels following the same trajectory. Also assume that the parcels lie on the same vortex line. • At some later time, the parcels have moved and the segment of the vortex line moves as a material line according to the trajectories of the parcels.

4. But what if the tilting is extremely abrupt, e.g., a strong gust front tilts vortex lines? Simpson, J.,1982: Cumulus rotation: Model and observations of a waterspout-bearing cloud system. Intense Atmospheric Vortices. L. Bengtsson and J. Lighthill, Editors. Springer-Verlag, 161–173.

5. Simulation of density current in environment having strong zonal and meridional shear • “Mother of all density currents”—5-km-deep block of cold air with a minimum q’ of –12 K • Extreme low-level shear/horizontal vorticity: (x0, h0, z0) = (–0.02, 0.02, 0.00) s–1 q’

6. Simulation of density current in environment having strong zonal and meridional shear streamlines • Just ahead of the almost vertical wall of the density current, warm air rises rapidly (w>20 ms–1 as low as 1 km above ground)

7. Simulation of density current in environment having strong zonal and meridional shear vertical velocity • Just ahead of the almost vertical wall of the density current, warm air rises rapidly (w>20 ms–1 as low as 1 km above ground)

8. Simulation of density current in environment having strong zonal and meridional shear vortex lines • Environmental vortex lines turn abruptly upward at the leading edge of the density current, but the peak z(0.02 s–1) is located well aloft at 3 km. • Despite the large environmental x in the lowest 1 km, the maximum zat 25 m (the lowest scalar level) in the warm air ahead of the density current is only 0.001 s–1.

9. Simulation of density current in environment having strong zonal and meridional shear vertical vorticity • Environmental vortex lines turn abruptly upward at the leading edge of the density current, but the peak z (0.02 s–1) is located well aloft at 3 km. • Despite the large environmental x in the lowest 1 km, the maximum z at 25 m (the lowest scalar level) in the warm air ahead of the density current is only 0.001 s–1.

10. Simulation of density current in environment having strong zonal and meridional shear pressure • The culprit: a stagnation high is present at the surface at the leading edge of the density current • Warm parcels encounter an adverse pressure gradient and decelerate as they get within about 2 km of the gust front. H

11. Simulation of density current in environment having strong zonal and meridional shear zonal vorticity • Parcels are compressed in the east-west direction (and stretched vertically to conserve mass). • The magnitude of xin the lowest 100 m decreases from 0.02 s–1 to 0.002 s–1 by the time the streamlines turn upward at the density current’s leading edge. • For a steady density current, it can be shown that x∝u and z∝w.

12. Conclusions • It cannot be argued that because there is large amount of horizontal vorticity in a surface-based layer in the environment, abrupt tilting of it at an “obstacle” (such as a gust front or topographical barrier) will produce similar strength vertical vorticity very close to the surface. • Linear thinking (i.e., assuming that horizontal vorticity is unmodified from environmental values) is misleading in this case because the abrupt tilting is unavoidably associated locally with a stagnation flow that greatly compresses the horizontal vortex tubes prior to tilting.

13. Summary • The results are consistent with the Dahl et al. (2012) trajectory study, which found that only outflow parcels entered the near-surface vortices of their simulated supercells Trajectories entering the vortex Trajectories originating in the inflow Dahl et al. (2012)

14. Point of clarification Rotunno & Klemp (1985) • Note, however, that in papers that have tracked material circuits, the circuits are much broader than the core radius of the near-surface mesocyclone (and in some studies the circuits are also several hundred meters above the surface) • These circuits extend into the inflow at the start of the backward trajectory calculations; thus, some of the parcels comprising the material circuits really do come from the inflow • i.e., just because a material circuit has large positive circulation at its starting point surrounding a mesocyclone doesn’t imply that every parcel comprising the material circuit has positive vorticity (or even contributes positively to the circulation) outflow inflow outflow inflow