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