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Vortex Sheets, Vortex Rings, and a Mesocyclone
David S. Nolan
Division of Meteorology and Physical OceanographyRosenstiel
School of Marine and Atmospheric Science
University of Miami
This work was supported by the Department of Energy
and the University of Miami.
I. What in the world is this about?
II. Three-dimensional vortex methods
III. Simulations of mesocyclone formationIV. Future work
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I. Motivation
Thunderstorm rotation and storm splitting are known to be
caused
by the tilting of low-level, horizontal vortex lines by
convective updrafts.
What is the simplest possible representation of this
process?
Perhaps, an updraft interacting with a low-level shear zone:
x
z
U(z)
w(x,y,z)
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Simplifying further, lets use and .
And, lets represent the flow using the simplest objects
possible.
The updraft becomes a column ofvortex rings, and the shear
zone
becomes a collection ofvortex sheets:
constant= 0=
x
y
z
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... and lets simulate these things using vortex methods:
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II. Three Dimensional Vortex Methods
Apurely Lagrangian approach to simulating fluid dynamics
In three dimensions, each vortex filament is represented by a
chain of segments:
The vorticity distribution around the segment remains fixed in
time;
Accuracy is achieved with sufficient, overlapping cores.
To maintain accuracy, each segment must be shorter than it is
wide.
As vortex lines stretch, new elements are added at the
midpoints.
= 0
> 0
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In general, the velocity field can be inverted from the
vorticity,
so that each endpoint moves according to the velocity induced by
allthe segments:
regularizes the singularity, and is related to the vorticity
distribution around
the segments.
Does this really work?
A large body of literature in mathematical fluid dynamics says:
yes!
u x( ) x x'( ) x'( )4 x x' 3
------------------------------------- x'dR
3
= =
u x( ) 14------ j
xjc
x( ) xj( )
xjc
x3
-------------------------------------- fxj
cx
----------------
j=
f r( )
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Initial Conditions for Unidirectional Shear
Low-level shear: 10 ms-1 over 1 km depth; Updraft: 11 ms-1.
Free-slip, impermeable lower boundary (w = 0).
Doubly-periodic in x andy, with a period of 8 km.
Boundary conditions are enforced using image vortices.
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Vel. Slice at y=0, t=0 maxvec=11.2393
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3D Vortex Lines at t=0 elements=731
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How does the flow evolve?
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3D Vortex Lines at t=60 elements=733
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3D Vortex Lines at t=180 elements=875
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3D Vortex Lines at t=300 elements=979
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3D Vortex Lines at t=420 elements=1083
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One trick: All vortex ringsare held fixed
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Counter-rotating updrafts are generated: (4 sheets, more
elements)
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Velocity at z=1500 at t=360 max [vx,vy]=6.38 vz=1.41 to 6.98
int=0.839
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Velocity at z=1500 at t=480 max [vx,vy]=7.11 vz=1.43 to 6.97
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Vort. z=750 t=360 maxvec=1.4e02 vz=6.0e03 to 6.0e03
int=1.2e03
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x 103
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Vort. z=750 t=480 maxvec=1.6e02 vz=7.7e03 to 7.6e03
int=1.5e03
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x 103
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Now, what if we rotate the vortex sheets with height?
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3D Vortex Lines at t=0 elements=731
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U
V
Hodograph
z = 0
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Now the lifting and stretching of the vortex lines is
asymmetric....
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3D Vortex Lines at t=60 elements=733
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3D Vortex Lines at t=180 elements=873
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3D Vortex Lines at t=300 elements=990
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3D Vortex Lines at t=420 elements=1105
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Now, the updrafts and the vortices are asymmetric!
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Velocity at z=1500 at t=360 max [vx,vy]=6.48 vz=1.41 to 7.82
int=0.923
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Velocity at z=1500 at t=480 max [vx,vy]=7.08 vz=1.39 to 7.98
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Vort. z=750 t=360 maxvec=1.1e02 vz=4.3e03 to 5.1e03
int=9.4e04
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Vort. z=750 t=480 maxvec=1.3e02 vz=5.9e03 to 6.9e03
int=1.3e03
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Some Conclusions...
The formation of rotating updrafts can be simulated
with inviscid, incompressible fluid flow and 3-D vortex
methods
The rotation with height of the low-level shear vector is
sufficient to generate
stronger updrafts and stronger low-level vorticity in one of the
mesocyclones
Tornado formation ... ?
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3D Vortex Lines at t=480 elements=2931
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3D Vortex Lines at t=480 elements=3250
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