Complex lamellar vector fields

In this final section, it is shown that the three magnetic field components of electromagnetic radiation in 0(3) electrodynamics are Beltrami vector fields, illustrating the fact that conventional Maxwell-Heaviside electrodynamics are incomplete. Therefore Beltrami electrodynamics can be regarded as foundational, structuring the vacuum fields of nature, and extending the point of view of Heaviside, who reduced the original Maxwell equations to their presently accepted textbook form. In this section, transverse plane waves are shown to be solenoidal, complex lamellar, and Beltrami, and to obey the Beltrami equation, of which B is an identically nonzero solution. In the Beltrami electrodynamics, therefore, the existence of the transverse 1 = implies that of , as in 0(3) electrodynamics. 

As argued by Reed [4], the Beltrami vector field originated in hydrodynamics and is force-free. It is one of the three basic types of field solenoidal, complex lamellar, and Beltrami. These vector fields originated in hydrodynamics and describe the properties of the velocity field, flux or streamline, v, and the vorticity V x v. The Beltrami field is also a Magnus force free fluid flow and is expressed in hydrodynamics as... 

Now, if a Beltrami field is simultaneously complex lamellar, (1) combined with (3), then curl v is both perpendicular and parallel to v. This can happen only if curl v is zero (that is, the field v is curl-less, or lamellar). Hence a vector field that is simultaneously a complex lamellar and a Beltrami field is necessarily lamellar. If the divergence of (5) is taken, we obtain... 

There are two known standard methods for decomposition of any smooth (differentiable) vector field. One is that attributed to Helmholtz, which splits any vector field into a lamellar (curl-free) component, and a solenoidal (divergenceless) component. The second, which divides a general vector field into lamellar and complex lamellar parts, is that popularized by Monge. However, the relatively recent discovery by Moses [7] shows that any smooth vector field— general or with restraints to be determined—may also be separable into circularly polarized vectors. Furthermore, this third method simplifies the otherwise difficult analysis of three-dimensional classical flow fields. The Beltrami flow field, which has a natural chiral structure, is particularly amenable to this type analysis. 

---