Download Roles of EHD, MHD, and (T)HD in Tornadic Thunderstorms

Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
Roles of EHD, MHD, and (T)HD in Tornadic Thunderstorms
H. Kikuchi
Institute for Environmental Electromagnetics
3-8-18, Komagome, Toshima-ku, Tokyo 170, Japan
e-mail: [email protected]
Abstract
In many cases, tornadoes are thought to be composed of uncharged and charged components
different from each other in terms of velocity, vorticity, helicity, and appearance (shape and luminosity). Their usual visible dark portion may correspond to uncharged tornadoes, while lu- minous
or bright part may involve charged tornadoes, accompanying lightning discharge with return strokes.
Usually, visible tornadoes have been considered to be ascending hot streams of thermo-hydrodynamic
origin. It is, however, newly claimed that quadrupole-like cloud-charge configurations with their
images onto ground can be a source-origin of helicity and vortex gen- eration in large-scale even if
ascending fluids are uncharged as well as small-scale helical tur- bulence, implying the possibility of
additional large-scale vortex generation..It is shown that tornadic thunderstorms can be now described
by a new ‘electrohydrodynamics (EHD)’ with novel physical concepts of ‘electric reconnection’ and
‘critical velocity’. Actually, artificial mapping of charge distributions and electric field lines onto
existing fluid vortex lines sketched for Minneapolis, Minesota tornado provides a reason- able
self-consistent overall picture of the EHD model with particular reference to tornadic destructtion by
dust cluster injection into vor- tex breakdown or electric cusp (X-type and/or O- type) points.
1. Introduction
While single uncharged tornadoes have extensively been investigated on the basis of (thermo-)hydrodynamics ((T)HD), no unified view or theory of uncharged and charged hybrid tor- nadoes
based on quadrupole-like cloud-charge configurations with their images onto ground has been
attempted as yet, though observations have been fairly accumulated, but now can be described by a
new ‘electrohydrodynamics (EHD)’ with novel physical concepts of ‘electric reconnection’ and
‘critical velocity’ [1]. In tornadic thunderstorms, fluid vortex and electric field transport becomes
significant, since earth’s magnetic field could be ignored, though mag- netic field generation due to
return stroke currents can be drawn by magnetohydrodynamics (MHD). During a lightning stroke,
there should occur two kinds of uncharged and charged flows, and electric field lines tend to coincide
with fluid vortex field lines, since both transport equations may satisfy the Kelvin-Helmholtz equation
due to possible high Reynolds and elec- tric Reynolds numbers. This may occur at least during an
initial stage of return stroke, indicat- ing that fluid vortex breakdown points also tend to merge electric
cusps, X-type and O-type with the addition of large-scale vortex generation due to EHD helical
turbulence. These results are applied to Minneapolis, Minesota tornado [2] in reference to tornado
destruction by dust cluster injection into vortex breakdown or electric cusp (X-type and/or O-type)
points
2. Equations of Fluid Vorticity, Magnetic, and Electric Transport
and the Kelvin-Helmholtz theorem
∂Ω
1
 = ∇ × (v × Ω) + ν∆Ω + ∇ ×  (qE + J × B),
∂t
ρ
(1)
where Ω = ∇ × v is the fluid vorticity, ρ and v the density and velocity of fluid, ν the kinetic
viscosity, q the charge density, E the electric field, B the magnetic field, and J is the current density.
The first, second, and third terms on the right-hand side of Eq.(1) represent he con- vection of the fluid,
the diffusion of vortex lines due to viscosity, and the production and trans- port of vorticity due to
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Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
space charges and currents, respectively.
∂B
η ∂2B
η∂
 = ∇ × (v × B) + η∆Β − 2 +  {∇ × (v × B)
∂t
cD ∂t2 cD2∂t
µ
1
−  ∇ × (v × H)} +ηµ∇ × qv + η{µ∆(v × D) − ∆(v
× E)},
n2
cD2
(2)
where η = 1/σµ is the magnetic diffusivity, µ the permeability, cD2 = c2/n2 = 1/εµ, c2 = 1/ε0µ0 , ε the
dielectric constantant, n the refractive index, and D is the electric displacement. The terms on the
right-hand side represent the convection (first and forth), diffusion (second), radiation (third), and
production (fifth and sixth) of the magnetic field. It is noted that the magnetic field is produced by
space-charge convection currents and electric fields.
∂2D
∂
∂J
e ∂

= ∇ × (v × D) + cD2(∆D − ∇q) −  − 2  ∇ × (v × E)
2
∂t
∂t
∂t
n ∂t
1
1
+ {∇∇·(v × B) − ∆(v ×B)}− {∇∇·(v
×H) −∆(v ×H)}
µ
n2
(3)
The terms on the right-hand side represent the convection (first and fourth), radiation (second), and
production (third, fifth, and sixth) of the electric field. It is noted that the electric field is produced by
conduction and space-charge convection currents and magnetic fields.
While fluid vorticity and magnetic field transport are characterized by the Reynolds number, R =
vL/ν (L: the characteristic length) and the magnetic Reynolds number, RM = vL/η = σµvL (σ: the
conductivity) respectively, electric field transport is also characterized by a new electric Reynolds
number’ defined as RE =εµvL/T (T: the characteristic time).
When these three kinds of Reynolds numbers are high enough, namely R » 1, RM » 1, RE »1,
Eqs. (1)~(3) are all reduced to the Kelvin-Helmholtz equation:
∂Ω
 = ∇ × (v × Ω),
∂t
∂B
 = ∇ × (v × B),
∂t
∂D
 + q v = ∇ × (v × D),
∂t
(4)
where the third of Eqs.(4) represent the space-charge related Kelvin-Helmholtz theorem, indi- cating
that the displacement current is added by the space-charge current for electric field trans- port. This
implies that the space-charge related frozen-in field concept holds rather than the conventional
source-free frozen-in field concept for fluid vortivity and magnetic field transport.
3. EHD vortices in an Electric Quadrupole and in EHD Helical Turbulence
Electrodynamics of an uncharged particle and EHD of an uncharged fluid in an electric quadrupole
represent a new type of helical motion of the particle and EHD vortex generation due to polarization
effects on the particle and .fluid and have been developed extensively [1].
While quarupole-associated helicity or vortex generation is one of most important features of EHD
vortices, the presence of EHD helical turbulence on the background.could be addition- al source-origin
of large-scale EHD vortex generation. Such EHD turbulence in electric, mag- netic, and space-charge
fields in addition to the density and velocity could be created, for in- stance, by the unsymmetry of an
external DC electric field. A new equation of EHD charged vortices is derived by means of a
functional technique with variational derivatives (functional and the random-force method) as follows:
∂Ω
1
1
 − ηn(0)Ω −  [g(0) − 2n(0)]∇ × Ω = ν∆Ω,
∂t
2
2
832
(5)
Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
where Ω = ∇ × <v> is the vorticity, η the ratio of charge to mass density, g(0) and n(0) are the fluid
and electric turbulent helicity coefficients.
4. Roles of HD, MHD, and EHD in Tornadic Thunderstorms
and Relevance of EHD to them
Basically the fluid or environment in the HD regime is a non-ionized fluid, while MHD and EHD
flows occur in a conducting and a dielectric or semiconducting fluid, respectively.
While tornadoes are, in many cases, thought to be composed of uncharged and charged components different from each other in terms of velocity, vorticity, helicity, and appearance such as shape
and luminosity. Their usual visible dark portion may correspond to uncharged torna- does, while
luminous or bright part may involve charged tornadoes, accompanying lightning discharge with return
strokes. Usually, visible tornadoes have been considered to be ascending hot streams of
thermo-hydrodynamic origin. In many cases. however, a quadrupole-like cloud- charge configurations
with their images onto ground can be a source-origin of helicity and vor- tex generation in large-scale
even if ascending fluids are uncharged as well as small-scale heli- cal turbulence, implying the
possibility of additional large-scale vortex generation. Usually horizontal or funnel scales and vertical
scales or heights of bottom cloud are considered as l ~ 10−100 m, h ~ 50−500 m respectively, forming
an ascending uncharged tornado. Its estimated vertical wind speeds, 20~200 m/s seem to well agree
with observations. In contrast, vertical wind speeds of charged tornadoes are thought to be much faster
and to be the speed of return stroke, say one third the velocity of light.
5. Two Kinds of Self-Organization in EHD Vortices and Application
to Tornado Destruction by Dust Cluster Injection
As already suggested in Sec.3, the study of charged EHD vortices as one of self-organiza- tional
processes of large-scale coherent structure generation in helical turbulence reveals that:
(1) Space charges with helical electric and magnetic field turbulence can be additional factor for
symmetry-breaking, leading to the generation of large-scale vortex structures by pumping out some of
the energy of helical turbulence into large-scale structures;
(2) The presence of an external DC electric field within a turbulent background can be another
symmetry-breaking factor as an origin of vortex-dynamo;
(3) Both space charges and a background DC electric field can cause the increase of the growth rate of
large-scale structures;, namely the growth rate of EHD vortices is larger than that of HD vortices.
It is newly suggested that in EHD vortices, a fluid vortex breakdown or merging point and electric
cuap or reconnection point observed from tornadic thunderstorms such as Minneapolis, Minesota
tornado might be a manifestation of self-organization to coalescence of fluid vortex and electric
(displacement) field lines (Fig.1)[2]. As already suggested in Sec.2, the study based on this principle
reveals that:
(1) The Kelvin-Helmholtz theorem and frozen-in field concept indicate that a vortex line
which initially coincides with an electric (displacement field line continues to do for sufficient- ly high
fluid and electric Reynolds numbers, providing self-organization to coalescence of flu- id vortex and
electric (displacement) field lines;
(2) When the fluid vortex or electric field constitutes a stagnation point or electric cusp, respectively, the concept of coalescence of fluid vortex and electric (displacement) field lines for
sufficiently high fluid and electric Reynolds numbers indicates that a stagnation point and an electric
cusp must coincide at one point. However, any perturbation exerted on a region of fluid stagnation or
an electric cusp, for instance invasion of a dust particle or cluster, can cause a lo- cal decrease in fluid
and electric Reynolds numbers, leading to a vortex line merging and electric electric cusp must
coincide at one point. However, any perturbation exerted on a region of fluid stagnation or an electric
cusp, for instance invasion of a dust particle or cluster, can cause a lo- cal decrease in fluid and electric
Reynolds numbers, leading to a vortex line merging and electric electric cusp must coincide at one
point. However, any perturbation exerted on a region of fluid
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Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
Figure.1. Relation between charge distributions, vortex and electric field lines for charged tornadoes: mapping of electric
field lines onto vortex lines. Summary cross-sectional sketch for vortex lines has been adopted from conjec- tured flow
structure of Minneapolis, Minnesota, tornado (after Pauley and Snow [1988]. An artificial mapping of electric charge
distributions and electric field lines onto existing fluid vortex lines. In addition, figure also indicates a method of tornado
destruction by dust cluster injection into vortex breakdown or electric cusp (X-type) point.
stagnation or an electric cusp, for instance invasion of a dust particle or cluster, can cause a lo- cal
decrease in fluid and electric Reynolds numbers, leading to a vortex line merging and elec- tric except
for this singular region, the stagnation and electric cusp also coincide, therefore virtually exhibiting
overall coalescence of fluid vortex and electric (displacement) field lines including fluid vortex
breakdown or merging point and electric reconnection point, X-type or O-type;
Finally, the results thus obtained for EHD vortices are applied to a typical Minneapolis, Minesota
tornado event in reference to tornado destruction. Fig.1 shows relation between charge distributions,
vortex and electric field lines for charged tornadoes: mapping of electric field lines onto vortex lines.
An artificial mapping of electric charge distributions and electric field lines onto existing fluid vortex
lines could produce a reasonable consistent charge distri- bution on thunderclouds that is thought most
likely to occur from observations. In addition, figure also indicates a method of tornado destruction by
dust cluster injection into vortex breakdown or electric cusp (X-type and/or O-type) points.
REFERENCES
1. H. Kikuchi, Electrohydrodynamics in Dusty and Dirty Plasmas, Kluwer Academic Publishers,
Dordrecht/The Netherlands, 2001.
2. R.L. Pauley and J.T. Snow, Mon. Weather Rev. 116, 2731, 1988.
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