A quaternionic structure in the three-dimensional Euler and ideal magneto-hydrodynamics equations

By considering the three-dimensional incompressible Euler equations, a 4-vector ζ is constructed out of a combination of scalar and vector products of the vorticity ω and the vortex stretching vector ω·∇u=Sω. The evolution equation for ζ can then be cast naturally into a quaternionic Riccati equation. This is easily transformed into a quaternionic zero-eigenvalue Schrödinger equation whose potential depends on the Hessian matrix of the pressure. With minor modifications, this system can alternatively be written in complex notation. An infinite set of solutions of scalar zero-eigenvalue Schrödinger equations has been found by Adler and Moser, which are discussed in the context of the present problem. Similarly, when the equations for ideal magneto-hydrodynamics (MHD) are written in Elsasser variables, a pair of 4-vectors ζ± are governed by coupled quaternionic Riccati equations.