## Zero Magnus Force

They are called “force free” because the vorticity (curl v) and velocity (v) vectors are parallel, so there is no Magnus force as described in Reed Section 5. As described by Reed on his page 538 similar Lorentz force free structures exist in electrodynamics. I agree that the solution of the free field equations is given by multiplying the Beltrami solution by exp (i omega t). The Beltrami solution may contain exp (- i kappa Z), so multiplying the two gives exp (i (omega t – kappa Z)).

To: EMyrone@aol.com
Sent: 28/02/2014 09:21:42 GMT Standard Time
Subj: Beltrami Solutions

In the appendix of the Marsh book, eq. A1.1 could be of intereset, but this seems to be a plane wave propagating in Z direction which we have already considered.

Two other points perhaps deserve discussion:
1. Beltrami fields are called force-free fields. What is the background of this statement? E and B are “force fields”, a test charge in such a field experiences a force, even in a Beltrami field. Only the vacuum can be force-free (potential without E/B fields).

2. As Marsh points out, multiplying a vector field by a factor exp(i omega t) describes a standing wave. So Beltrami fields defined in this way are standing waves. On the other hand we have considered plane waves with the factor exp( i (omega t – kappa Z)). These are not standing waves. Obviously both types of waves can constitute a Beltrami field. In particular it is possible to generate standing waves at “one end”, in mechanics you need two fixed ends to obtain a standing wave. Is this true and an important result?

Horst

EMyrone@aol.com hat am 27. Februar 2014 um 19:55 geschrieben:

Good point, the previous literature may well be wrong, yet again.

In a message dated 27/02/2014 14:59:15 GMT Standard Time, mail@horst-eckardt.de writes:

PS: as shown in paper 257 the “chaotic solution” is only a Beltrami field for A=B=C.

Horst

EMyrone@aol.com hat am 27. Februar 2014 um 11:00 geschrieben:

It would be very interesting to animate these solutions. The first one is the most general solution (3) already graphed in stills by Horst Eckardt, where a is any constant vector and where psi is a scalar solution of the Helmholtz wave equation. In general psi is involves the spherical harmonics, and it possible that Maxima and Mathematica provide tables of solutions of the Helmholtz equation. Reed, Marsh and Wikipedia do not give sufficient information about the complete solution. The Reed solution already animated is a cylindrically symmetric solution. There are very many solutions possible and all are possible solutions of the free field ECE equations which used to be known as the Maxwell Heaviside equations. In this note the general solution is reduced to a plane wave solution. In fact it would be interesting to animate the plane waves and B(3). Secondly the Lundquist type solution is given in eqs. (26) to (28). It would be interesting to check that these equations are actually correct, then animate them. Thirdly a chaotic solution is given in eqs. (29) to (31). The relevant references to Marsh are given. All if this refutes Higgs boson theory entirely, and the large cuts to particle physics mean that legislators are beginning to smell a bad kipper on Friday afternoon in Swansea market, a kipper known as a boson.