Trkalian field

A decomposition of a Trkalian field into Fourier modes... 

Let us start with the Trkalian field A as defined above, with its time-like component ... 

However, the principle of field superposition is verified if the condition k = is imposed. On the other hand, the superposition of two Trkalian fields generates a... 

As a consequence, the gauge transform written above have the property to turn an ordinary field A into a Trkalian field Ao. 

However, the preceding conditions, as opposed to what usually happens in the real case, are not sufficient conditions to define a complex Trkalian field. Indeed, the couple of authors we referred to, demonstrated that for 2 / 0 and / / 0, the complex field Bc is an ordinary field if we chose Bc = ejIIVF, where F is a complex quantity having the expression F = ( + ) 7 . This results from the equality VA Bc = /VIIABc which implies Bc VA Bc = kB2c = 0 where is a constant different from zero. In this case, the definition of the Bc field implies the following relations ... 

In the case of uniform c(grad c = 0), curl v = w (vorticity) is also a Beltrami field, possessing the same coefficient c. This type of vector field is called a Trkalian field, after Trkal, a Russian researcher who studied Beltrami flows... 

Since the left side of (14) is necessarily positive, we conclude that a Beltrami field v and the curl of its curl [right side of (14)] always meet at an acute angle. If the angle is zero, then curl v is also a Beltrami field. In this last case, because of considerations above, v is a Trkalian field. 

A divergenceless magnetic field (div B = 0) is assumed in most classical electromagnetic field applications. Thus, this constraint is usually applied in the Beltrami FFMF condition (21) when determining solutions to this equation. From previous considerations, the solenoidal FFMF describes a Trkalian field... 

In this application we consider EM fields in free space consequently both E and H are solenoidal and satisfy Trkalian field relations. Thus, taking the curl of (71), both vector fields satisfy Helmholtz vector wave equations ... 

As is well known, this transformation has the form of a duality transformation [66-68], where, in the above special case, the vectors Aj and A2, similar to E and H, are compelled to satisfy Trkalian field relations ... 

---