Vortex knots for the spheromak fluid flow and their moduli spaces

Abstract

New exact solutions to the Euler hydrodynamics equations are constructed. A method for the study of vortex knots is developed for a special class of ideal fluid flows – the axisymmetric ones satisfying the Beltrami equation curlV(x)=λV(x). The method is based on a construction of the moduli spaces of vortex knots S(R3). Applying the method to the spheromak fluid flow we demonstrate that only those torus knots Kp,q are realized as vortex knots for which p/q belongs to the interval I1:0.5<τ<M1≈0.8252. We prove that each torus knot Kp,q with 1/2<p/q<1/2 is realized on countably many invariant tori T2⊂R3, while torus knots with 1/2<p/q<M1 are realized only on finitely many tori. The moduli spaces of vortex knots Sm(Ba3) (m=1,2,⋯) are constructed for the spheromak fluid flows inside a ball Ba3 of radius a.