Very interesting! It would be especially important to see how this structure goes over into vacuum plane waves and B(3) or into a helicoidal vacuum structure for electromagnetism as sketched in to 257(2). In order to make them propagate along Z the phase omega t – kappa Z must be incorporated. It seems that Beltrami flow is of interest to geology and geography, and is thought to cause rivers to meander. Some of the Beltrami structures are very intricate and there is a transition to turbulence at some point.

Sent: 29/01/2014 15:35:23 GMT Standard Time

Subj: Re: Analytical Solutions of the Beltrami equationThe factor of 2 can be omitted. However B fulfils only the Beltrami equation with T1=T2=T3, see below. It is also interesting that B can be found by a differential equation.

EMyrone@aol.com hat am 29. Januar 2014 um 16:03 geschrieben:

Google “Helicoidal solution of the Beltrami Equation”, first site, “Generalized Helicity Conservation and Beltrami … ” gives many interesting solutions. For example:

B sub X = T2 sin 2TZ + T3 cos 2TY

B sub Y = T3 sin 2TX + T1 cos 2TZ

B sub Z = T1 sin 2TY + T2 cos 2TXwhere T1, T2 and T3 are arbitrary functions.

In a message dated 29/01/2014 14:19:17 GMT Standard Time, writes:

I just wrote to Myron that it is difficult to find an analytical example (in cylinder coordinates) for the helical structure in the figure of the Reed article. If we want to investigate such structures it probably cannot be reduced to 2D because of the rotation in Z direction. However a Beltrami equation without this property should be possible to be formulated in 2D spherical coordinates.

Horst

Doug Lindstrom hat am 29. Januar 2014 um 14:35 geschrieben:

There is a good chance that flexpde can handle it using complex variables. It would be worth a try using radial symmetry to reduce the problem to 2D. Doug

On Wednesday, January 29, 2014, Horst Eckardt < mail> wrote:

PS:

Doug, is there a chance to solve the Beltrami equation numerically by the FEM method? I guess it has to be in 3D.Horst

Horst Eckardt < mail@horst-eckardt.de> hat am 29. Januar 2014 um 08:42 geschrieben:

Myron, Doug,

The review article is good, nethertheless I have an understanding problem with interpretation of the Beltrami equation. There are interesting diagrams how such a flow can be imagined, but to my understanding the curl of a vector is always perpendicular to the vector itself. This seems also to be the case in the longitudinal flux field examples. How can the curl be parallel to the vector?Horst

EMyrone@aol.com hat am 28. Januar 2014 um 18:53 geschrieben:

Many thanks, this is an important review by Donald Reed.

To: EMyrone@aol.com

Sent: 28/01/2014 13:45:47 GMT Standard Time

Subj: Re: Fwd: Beltrami fields Forgot the link – NormanI’ll send the link again . I think this is from the Reed paper you refered to . Here is the bit on B3 Regards Norman.

“Evans/Vigier Longitudinal B(3) Field and Trkalian Vector Fields

Developed over the past decade, concurrently with both Hillion/Quinnez and Rodrigues/Vaz SU(2) EM field models, but based upon a different non-Abelian gauge group, is the so-called Evans/Vigier longitudinal B(3) field representation [89-93]. In this model, a Yang-Mills gauge field theory [94] with an internal O(3) gauge field symmetry [95] is invoked to account for various magneto-optical effects which are claimed to be a function of a third magnetic field vector component that has been termed B(3). O of the central theorems of O(3) electrodynamics is the B-cyclic theorem:

B(1) x B(2) = i B(o) B(3)*, (93)

A conjugate product which relates three basic magnetic field components in vacuo defined as:

B(1) = �(�)√2 ( � � + �)exp(� ∅) (93a)

B(2) = B(0)√2 (-i i + j)exp(-i ∅) (93b)

B(3) = B(0)k, (93c)

where ∅ = ωt − ��, a phase factor, and i, j, k are the three unit vectors in the direction of the axes x, y, and z, respectively. Although the existence of the B(3) field has been a subject of controversy both pro and con over recent years, Evans recently claimed [96] that these magnetic field components encompassed by the relations (93a-c), along with the electric field components as well as the components of the magnetic vector potential (A), are themselves components of a Beltrami-Trkalian vector field relations (assuming the Coulomb gauge div A = 0). This is readily verified in the case of (93ab), since they present the form of the circularly-polarized solution to the Moses eigenfunctions of the curl operator we have discussed formerly in connection with turbulence in fluid dynamics.

Associated with the above developments, is the increasing importance given to hypercomplex formalisms for modeling the symmetries in elementary particle physics and quantum vacuum morphology. As discussed in former papers [97,98], the author believe that the most appropriate algebra for describing a hypothesized vortical structure for quantum-level singularities, as well as their macroscopic counterparts (Beltrami-type fields), is the biquaternion algebra (hypercomplex numbers of order 8) – the Clifford algebra of order 3, represented by the Pauli algebra Cl(3,0) such as previously examined in the Rodrigues/Vaz model. For instance, it is known that in a macroscopic Euclidean context, biquaternions are required to describe the kinematics/dynamics for the most general twisting (screw) movement of a rigid body in space [99,100]. It is therefore suggested that the most suitable formalism for screw-type EM fields of the Beltrami variety should transcend a traditional vectorial treatment, encompassing a para-vectorial hypercomplex formalism akin to the Clifford (Dirac) algebra used effectively to describe the electron spin in a relativistic context [101].

It is a conclusion in this regard that the founders of vector field analysis were remiss in failing to take into consideration account of the significance of the Beltrami field topology in addition to the traditional solenoidal, lamellar and complex-lamellar fields. An inclusion of the thorough examination of the Beltrami condition in the development of the vector calculus, would possibly have brought attention to the important intimate association of this field configuration with non-Abelian mathematical structure. If the history of vector analysis had taken this path, it is possible that the architects of vector field theory and classical electrodynamics, would not have been so quick to indiscriminately sever its connection from the natural quaternion-based foundation. Perhaps the recent work by Hillion/Quinnez, Rodrigues/Vaz, Evans, etc. [102,103] showing the necessity of considering non-Abelian models in electromagnetism, will be instrumental in helping to set the future of classical EM theory and vector field theory in general, on a firmer foundation.”http://vixra.org/pdf/1207.0080v1.pdf

On 1/28/2014 3:38 AM, EMyrone@aol.com wrote:

On 28/01/2014, at 7:00 PM, EMyrone@aol.com wrote:To Norman Page: I tried this link but I got "requested resource not available". Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, 't Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn't allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemis try, pe ace, literature, medicine and economics. What use is a non existent boson? From: norpag@att.net To: EMyrone@aol.com Sent: 27/01/2014 18:11:09 GMT Standard Time Subj: Beltrami fields Forgot the link - Norman http://vixra.org/pdf/1207.0080v1.pdfOn 28/01/2014, at 7:00 PM, EMyrone@aol.com wrote:

To Norman Page:

I tried this link but I got “requested resource not available”. Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, ‘t Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn’t allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemistry, peace, literature, medicine and economics. What use is a non existent boson?