1 / 22

Εugenio Beltrami, 16 November 1835 - 4 June 1899) “Considerations in Hydrodynamics” (1889) ‏

BELTRAMI FIELDS IN ELECTROMAGNETISM Theophanes Raptis 2009 Computational Applications Group Division of Applied Technologies NCSR Demokritos, Ag. Paraskevi, Attiki, 151 35. Εugenio Beltrami, 16 November 1835 - 4 June 1899) “Considerations in Hydrodynamics” (1889) ‏

newman
Download Presentation

Εugenio Beltrami, 16 November 1835 - 4 June 1899) “Considerations in Hydrodynamics” (1889) ‏

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. BELTRAMI FIELDS IN ELECTROMAGNETISMTheophanes Raptis2009Computational Applications GroupDivision of Applied TechnologiesNCSR Demokritos, Ag. Paraskevi, Attiki, 151 35

  2. Εugenio Beltrami, 16 November 1835 - 4 June 1899) • “Considerations in Hydrodynamics” (1889)‏ • Vorticity in Navier-Stokes eq. w = curlv • Mangus Flowv 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

  3. 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.

  4. 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 BIRKELAND CURRENTS

  5. 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

  6. Linear case: Equivalent with a special class of Helmholtz solutions • Leads to Chandrasekhar-Kendall eigen-functions. • Non-linear case: no known general solution

  7. 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 ]

  8. [Brownstein 1986] Equivalent to 4-wave interference – 2 pairs of “phase conjugated” waves PC

  9. 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)

  10. General solutions for the Spherical Beltrami problem [Papageorgiou-Raptis 2009 CHAOS conf.] • Introduce Vector Spherical Harmonics • Expansion of (1-A) leads to

  11. Equivalent to a “lossless” Transmission Line • Propagation condition • Evanescence • Hidden Lorentz Group 

  12. 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”

  13. Example : • leads to the system • where and s = +1. • The permutation holds for s = -1.

  14. Beltrami-TEM Waves • CASE I: • CASE II: (Dual Beltrami-Ballabh waves)

  15. For case 1 just replace • From previous example • Momentum transfer (<g> divergent!) • Angular Momentum

  16. 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

  17. 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

  18. Modulator Types (Simulations in Plasma UCLA-BPPL) Helical fields Sun Magnetic Field Due to Rotation

  19. A POSSIBLE “WARP” MODULATOR Local evolution : For E // B : Conformal Inversion of Lundqvist solution : Φ-Antenna Ζ-Coils

  20. 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

  21. A Roadmap for Gravito-Electromagnetism [Robert Forward 1963] Constitutive Relations in Curved metrics + linearized Field equations (Ramos 2006) TRY OPTICAL FIBERS?

  22. 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…

More Related