Velocity Beltrami vector fields

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... 

However, in a Beltrami field, the vorticity and velocity vectors are parallel or antiparallel, resulting in a zero Magnus force. The Beltrami condition (1) is therefore an equivalent way of characterizing a force-free flow situation, and vice versa. 

Figure 8. Dissected diagram of the vector configuration of a pair of Beltrami vortex filaments formed in the current sheath of the plasma focus (v — flow velocity, B = local magnetic field, j = current density, (>) — vorticity, Bo — background magnetic field).

This is clearly a Beltrami equation, but what is more amazing is that the field result (88) describes a solution to the free-space Maxwell equations that, in contrast to standard PWS, the electric (E0) and magnetic (Bo) vectors are parallel [e.g., Eo x Bo = 0, where Eo x Bo = i(E0 A Bo)], the signal (group) velocity of the wave is subluminal (v < c), the field invariants are non-null, and as (91) clearly shows, this wave is not transverse but possesses longitudinal components. Moreover, Rodrigues and Vaz found similar solutions to the free-space Maxwell equations that describe a superluminal (v > c) situation [71]. 

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