BELTRAMI FIELDS IN ELECTROMAGNETISM Theophanes Raptis 2009 Computational Applications Group Division...
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Transcript of BELTRAMI FIELDS IN ELECTROMAGNETISM Theophanes Raptis 2009 Computational Applications Group Division...
BELTRAMI FIELDS IN ELECTROMAGNETISM
Theophanes Raptis2009
Computational Applications GroupDivision of Applied Technologies
NCSR Demokritos, Ag. Paraskevi, Attiki, 151 35
Εugenio Beltrami, 16 November 1835 - 4 June 1899)
“Considerations in Hydrodynamics” (1889) Vorticity in Navier-Stokes eq. w = curlvMangus Flow v x (curlv) = 0 (force-free!)Three basic velocity field types• Solenoidal divv = 0• Lamellar v(gradv) = 0• Beltrami 2v(gradv)=grad|v|2, curlv = λv
Eigenvorticity : λ = v(curlv)/|v|2 = w(curlw)/|w|2
Quasi-static space magnetic fields
• [Lundquist 1951, Lust-Schluter 1954, Chandrasekhar 1957-1959]
• Relaxed state of plasma (from Force-Free condition)
• λ usually assumed constant
• If displacement current taken into account then exponential relaxation to equilibrium state.
BB
BJBB
0)(
BIRKELAND CURRENTS
Jovian Currents with the characteristic helical form
[K. Birkeland 1903, H. Alfven 1939, Dressler &Freeman, 1969, Navy Sat TRIAD – Zmuda & Armstrong, 1974]
Lundquist solution w. const. λ
)](),(,0[ 01 rJrJ B
The Generic Beltrami Problem
(1-A)
from
either(1-B)
or(1-C)
In case of const. λ we have a linear (Trkalian) flow [V. Trkal, 1910]In case of (1-B) we have a natural orthogonal frame
ArA ),( t
0)( AAA
00 AA
AA ,,
AA 1t
• Linear case: Equivalent with a special class of Helmholtz solutions
• Leads to Chandrasekhar-Kendall eigen-functions.
• Non-linear case: no known general solution
0)( 20
2 B
BBr ))(( 22
The paradox of parallel E & B fields
If one starts with a vector potential of the form
where φ is a solution of the scalar wave equation then one gets
[Chu & Okhawa ]
)()( nnA
ABAiEAA ,,
tkzkzB
tkzkzE
k z
cos]0,cos,[sin
sin]0,cos,[sin
],0,0[,0
0
0
B
E
kj
[Brownstein 1986]
Equivalent to 4-wave interference – 2 pairs of “phase conjugated” waves
PC
4
1
4
1 2
1,
2
1
ii
ii BBEE
)sin(),sin(
)cos(),cos(
4,34,3
2,12,1
tkzkatkzka
tkzkatkzka
iBjE
jBiE
Maxwell fields as complex Beltrami fields
[Silberstein 1907, Chubykalo 80’s, Lakhtakia 80s-90s, Hillion 90’s]
• Introduce the new vectors• Rewrite Maxwell equations• Monochromatic waves • Introduce Debye-Hertz potentials
• Beltrami condition acts like a filter on a primordial longtitudinal complex field (C = conj. operator)
iBEF FiF tn
FF n
tLMF *
rLMrLi ,,0, 22 t
001
10
tt
Cn
C
L
M
L
M
General solutions for the Spherical Beltrami problem
[Papageorgiou-Raptis 2009 CHAOS conf.]
• Introduce Vector Spherical Harmonics
• Expansion of (1-A) leads to lmlmlm r rNMrL ,,
0
0
)1(
lmlmlm
lm
lmlm
lm
lmlm
cr
a
r
bb
br
cc
acr
ll
• Equivalent to a “lossless” Transmission Line
• Propagation condition• Evanescence• Hidden Lorentz Group
YVVdr
dIrbI
ZIIdr
dVrcV
jj
j
2 )1(, 222
22 llL
r
L
)1(
||02
llror
)1(
||02
llror
0)()(2
222
r
Lrr 0)( 2222 LTYXs
Solutions w. special geometry (Rules of another game)
• Introduce partial vector fields
Utilize the natural frame where
is a field complementary to A.
• Naturally
• This also carries an apparent “charge”
)(,0 iji xyAA
,, CAA AA C
2|| A
AA C
)(2ix
• Example :
• leads to the system
• where and s = +1.
• The permutation holds for s = -1.
AA )(r
)()( rdrrh
1221 , yyyy
))(()(
))(()(
2211
1122
yrcrsyrc
yrcrsyrc
)](),(,0[)(sin 22111 rycryc A
))](sinh()),(cosh(,0[sin
1
))](sin()),(cos(,0[sin
1
21
21
rhcrhcr
rhcrhcr
A
A
Beltrami-TEM Waves
• CASE I:
• CASE II:
(Dual Beltrami-Ballabh waves)
BEBB
EBEE
)(,)(
)(,)(
uu
uu
MtM
t
EEEB
EBBE
)(,)(
)(,)(
uu
uu
MtM
t
• For case 1 just replace
• From previous example
• Momentum transfer
(<g> divergent!)
• Angular Momentum
sin
))(sin(sin
))(cos(
0
r
ctrhB
r
ctrhB
Br
MEi ctxu ,
0 grL
sin
))(cos(sin
))(sin(
0
r
ctrhE
r
ctrhE
Er
)2cos(sin
ˆ22
hr
rBEg
Can there be Zero-momentum waves?
Let there be 2 normal vector potentials on the sphere such that
so that
Then either or would cause
AAA
AAA
fr
fr
)(
)(22
22
),()(),(
),()(),(
AA
AA
r
r
AA AA
!0,),(// 2 LgAABE f
MACROSCOPIC HELICITY MODULATION
[Moffat 1969] Total Helicity Conservation
Gauge Invariant def.
Local Helicity Density fluctuations MUST propagate
TW = Twisting number, WR = Writhing number,
L = Linking number, NL,R = Left-Right Pol. Photons
MFdVdVH 000 ,BBAAB
rwRL WTLNNL ,2h
Modulator Types (Simulations in Plasma UCLA-BPPL)
Helical fields
Sun Magnetic Field
Due to Rotation
A POSSIBLE “WARP” MODULATOR
Local evolution :
For E // B :
Conformal Inversion of Lundqvist solution :
SSVt BABEh2
1
22 |||| ABhh EtMt
)](),(,0[
,,1
22
202
0
ryr
ryr
ru
rur
z
A
yyu
yuyu
zu
zu
02
0
)(
)(1
Ζ-Coils
Φ-Antenna
Sources for Helical Poynting Flux
Limiting cases:
• J=0 Parallel E – B fields
• E=0 force-free field
From we find
For we get
For Br = [0,y1(r),y2(r)] we approximate a helical flux
jiEBiE
JEiBB
ccc
c
2
2
B
BJiE
BBJ
cc
n)(
2
22
rJ ˆ)(rf BJg
A Roadmap for Gravito-Electromagnetism
[Robert Forward 1963]
Constitutive Relations in
Curved metrics
+ linearized Field equations
(Ramos 2006) TRY OPTICAL FIBERS?
00
0000
2/1
2/1
,h
hghh
hg
h
h
ii
EgHB
HgED
iiti
i
t
jkjktt
iikjkj
FhG
GcF
TtGcFcG
GttGcF
)(,
0
3/4
0,)(3/14
00
1
21
2
gg
Warp Engineeringvia Hopf Fibrations
[Ranada, Trueba 1996]
Geodetic Knot w. Hopf invariants
Problem : Can it fit the Alcubierre
Metric?
(Potentials must be velocity
Dependent, Spinning E/M fields?)
Fibers might have to become like…
22
*
22
*
||14,
||14
iiG
n
nnF