Can a gust front tilt horizontal
vortex lines to produce a tornado?
Paul Markowski
Pennsylvania State University
Bob Davies-Jones
Emeritus, NSSL
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.
A B
Hermann von Helmholtz
(1821–1894)
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.
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.
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’
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 m s–1
as low as 1 km above
ground)
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 m s–1
as low as 1 km above
ground)
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 z at 25 m (the lowest
scalar level) in the warm air
ahead of the density current is
only 0.001 s–1.
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.
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
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 x in 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.
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.
Summary
• The results are consistent with the Dahl et al. (2012) trajectory
study, which found that only outflow parcels entered the nearsurface vortices of their simulated supercells
Trajectories entering the vortex
Dahl et al. (2012)
Trajectories originating in the inflow
Point of clarification
• 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)
Rotunno & Klemp (1985)
outflow
inflow
outflow
inflow