BELTRAMI FIELDS IN ELECTROMAGNETISM

Theophanes Raptis2009

Computational Applications GroupDivision of Applied TechnologiesDivision 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 = curlv� Vorticity in Navier-Stokes eq. w = curlv

�Mangus 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|2Eigenvorticity : λ = v(curlv)/|v|2 = w(curlw)/|w|2

Quasi-static space magnetic fields

• [Lundquist 1951, Lust-Schluter 1954, Chandrasekhar

1957-1959]1957-1959]

• Relaxed state of plasma (from Force-Free condition)

• λ usually assumed constant

BB

BJBB

λ=×∇⇒

=×=××∇ 0)(

• λ usually assumed constant

• If displacement current taken into account then

exponential relaxation to equilibrium state.

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 λλ=B

The Generic Beltrami Problem

(1-A) ArA ),( tλ=×∇

from

either

(1-B)

or

(1-C)

0)( =∇+∇=∇ AAA λλλ

00 =∇⇔=∇ AA λ

AA λλ ∇=∇=Φ∂ −1

t

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

{ }AA ×∇∇ λλ,,

t

• Linear case: Equivalent with a special class of Helmholtz solutions

0)( 2

0

2 =+∇ Bλ

• Leads to Chandrasekhar-Kendall eigen-functions.

• Non-linear case: no known general solution• Non-linear case: no known general solution

BBr ×∇=+∇ λλ ))(( 22

The paradox of parallel E & B fields

If one starts with a vector potential of the form

)()( φφ nnA ×∇×∇+×∇=

where φ is a solution of the scalar wave equation then one gets

[Chu & Okhawa ]

)()( φφ nnA ×∇×∇+×∇=

ABAiEAA λωλ =−==×∇ ,,

[Chu & Okhawa ]

tkzkzB

tkzkzE

k z

ω

ω

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

i

i

i

i BBEE

)sin(),sin(

)cos(),cos(

4,34,3

2,12,1

tkzkatkzka

tkzkatkzka

ωω

ωω

mmm

mm

iBjE

jBiE

==

=±=

Maxwell fields as complex Beltrami fields

[Silberstein 1907, Chubykalo 80’s, Lakhtakia 80s-90s, Hillion 90’s]90s, Hillion 90’s]

• Introduce the new vectors

• Rewrite Maxwell equations

• Monochromatic waves

• Introduce Debye-Hertz potentials

iBEF ±=± µε±± ∂=×∇ FiF tn

±± =×∇ FF ωn

±±± ∂+= ϕϕ tLMF

*

( ) rLMrLi ×∇×∇=×∇=∇×−==∂−∇±= ,,0, 22 φφφφ

• Beltrami condition acts like a filter on a primordiallongtitudinal complex field (C = conj. operator)

( ) rLMrLi ×∇×∇=×∇=∇×−==∂−∇±= ±ΜΕ± ,,0, 22 φφφφ t

001

10=

∂

−

∂

×∇ ±φωtt

Cn

C

L

M

L

M

General solutions for the Spherical Beltrami problem

[Papageorgiou-Raptis 2009 CHAOS conf.]

• Introduce Vector Spherical Harmonics• Introduce Vector Spherical Harmonics

• Expansion of (1-A) leads to lmlmlm r Ψ=Ψ∇=Ψ∇×= rNMrL ,,

)1(=

+− lmlm ac

r

llλ

0

0

=−−+

=++

=−

lm

lmlm

lm

lm

lm

lm

lmlm

cr

a

r

bb

br

cc

acr

λ

λ

λ

&

&

• Equivalent to a “lossless” Transmission Line

dI

ZIIdr

dVrcV =−=→=

γ

λj

2 )1(, 22

2

22 +=−= llL

Lλγ

• Propagation condition

• Evanescence

• Hidden Lorentz Group

YVVdr

dIrbI =−=→=

λγj

j2 )1(,

2+=−= llL

rλγ

)1(

||02

+><

llror

λγ

)1(

||02

+<>

llror

λγ• Hidden Lorentz Group

�

)1( +ll

0)()(2

222 =−+

r

Lrr λγ 0)( 2222 =−+= LTYXs

Solutions w. special geometry (Rules of another game)

• Introduce partial vector fields

Utilize the natural frame where

)(,0 iji xyAA ==

{ }λ∇ ×∇= λCUtilize the natural frame where

is a field complementary to A.

• Naturally

• This also carries an apparent “charge”

{ }λ∇,, CAA AA ×∇= λC

2|| A

AAC×

=∇λ

• This also carries an apparent “charge”

)(2

ixρλ =∇

• Example :

• leads to the systemAA )(rλ±=×∇

))(()(

))(()(

2211

1122

yrcrsyrc

yrcrsyrc

λ

λ

=′

=′−

)](),(,0[)(sin 2211

1 rycryc−= θA

∫

))(()( 2211 yrcrsyrc λ=′

))](sinh()),(cosh(,0[sin

1

))](sin()),(cos(,0[sin

1

21

21

rhcrhcr

rhcrhcr

±=

±=

mθ

θ

A

A

• where and s = +1.

• The permutation holds for s = -1.

∫= )()( rdrrh λ

1221 , yyyy →−→

Beltrami-TEM Waves

• CASE I:

EBEE )(,)( uu t λλ =∂−=×∇ ΕΕ

• CASE II:

BEBB

EBEE

)(,)(

)(,)(

uu

uu

MtM

t

λλ

λλ

=∂=×∇

=∂−=×∇ ΕΕ

EBBE )(,)( uu t λλ =∂−=×∇ ΕΕ

(Dual Beltrami-Ballabh waves)

EEEB )(,)( uu MtM λλ =∂=×∇

• For case 1 just replace

• From previous example

0Br

−

=

MEi ctxu λλ =−→ ,

))(sin(

0

ctrh

Er

−

=

θ

θ

φ

θ

sin

))(sin(

sin

))(cos(

r

ctrhB

r

ctrhB

−−=

−=

θ

θ

φ

θ

sin

))(cos(

sin

))(sin(

r

ctrhE

r

ctrhE

−=

−=

r̂• Momentum transfer

(<g> divergent!)

• Angular Momentum 0=×∝ grL

)2cos(sin

ˆ22

hr θ

rBEg =×∝

Can there be Zero-momentum waves?

Let there be 2 normal vector potentials on the sphere such thatsphere such that

so that ( )( ) AAA

AAA

fr

fr

∝×∇=−∇

∝×∇=−∇ ⊥

λλ

λλ

)(

)(

22

22

),()(),(

),()(),(

ϕθλϕθϕθλϕθ

⊥⊥ =×∇

=×∇

AA

AA

r

r

Then either or would cause

( ) AAA fr ∝×∇=−∇ ⊥⊥ λλ )(22

⊥+ AA ⊥− AA

!0,),(// 2 ==+∝±∝ ⊥ LgAABE fλεµλ

MACROSCOPIC HELICITY MODULATION

[Moffat 1969] Total Helicity Conservation

Gauge Invariant def.

Local Helicity Density fluctuations MUST propagate

∫ ∫ ∇=−= MFdVdVH 000 ,BBAAB

WTLNNL +=−=Φ∝ ,2h

TW = Twisting number, WR = Writhing number,

L = Linking number, NL,R = Left-Right Pol. Photons

rwRL WTLNNL +=−=Φ∝ ,2h

Modulator Types (Simulations in Plasma UCLA-BPPL)

Helical fieldsHelical fields

Sun Magnetic FieldSun Magnetic Field

Due to Rotation

A POSSIBLE “WARP” MODULATOR

Local evolution :

SSVt >×<+>•<−∝∂ BABEh2

1

For E // B :

Conformal Inversion of Lundqvist solution :

><−>∝<−∝−∂=∂ 22 |||| ABhh λEtMt

,,1

2

02

0r

ur

ur

λλ

λεµλλ

=

−∝−→=→φ

λ

λ yuyu

zu

2

0)(1

=∂

=∂−

)](),(,0[ 22

ryr

ryr z

λλφ=A φλ yyu zu 0

2 )( =∂

Ζ-Coils

Φ-Antenna

Sources for Helical Poynting Flux

Limiting cases:jiEBiE

JEiBB

ccc

c

µωλ

µεωµω

εωλ

−−==×∇

+−==×∇

2

2

Limiting cases:

• J=0 Parallel E – B fields

• E=0 force-free field

From we find

ccc λ

B×∇×∇

BBJ λω

λ ×∇++=×∇c

n)(

2

22

For we get

For Br = [0,y1(r),y2(r)] we approximate a helical flux

( )BJiE λεω

−=c

c2

rJ ˆ)(rf≈ BJg ×∝

A Roadmap for Gravito-Electromagnetism

[Robert Forward 1963]

Constitutive Relations inConstitutive Relations in

Curved metrics

2/1

2/1

hg

h

h

+=

×+=

×+=−

−

µνµνµν ηEgHB

HgED

+ linearized Field equations

(Ramos 2006) TRY OPTICAL FIBERS?

00

0

000 ,h

hghh i

i ==r

µνµνµν

[ ]( )

ii

ti

i

t

jkjktt

iikjkj

FhG

GcF

TtGcFcG

GttGcF

)(,

0

3/4

0,)(3/14

00

1

21

2

gg ×∇=∂−−∂=

=∂+×∇

−∂−=∂−×∇

=∇+∂−∂=∇

−

−−

−

δπ

επ

Warp Engineeringvia Hopf Fibrations

( ) ( )22

*

22

*

||14,

||14 ζπ

ζζ

π +

∇×∇∝

+

∇×∇∝

iiG

n

nnF

[Ranada, Trueba 1996]

Geodetic Knot w. Hopf invariants

Problem : Can it fit the Alcubierre

Metric?

( ) ( )||14||14 ζππ ++ ii n

Metric?

(Potentials must be velocity

Dependent, Spinning E/M fields?)

Fibers might have to become like…