Beltrami Fields

Abstract

Solutions of the equation

$$ rot \overrightarrow B + \alpha \overrightarrow B = 0 $$

(9.1)

where α is a scalar function of space coordinates are known as Beltrami fields and are of fundamental importance in different branches of modern physics (see, e.g., [128], [82], [43], [125], [4], [55], [50], [67]). For simplicity, here we consider the real-valued proportionality factor α. and real-valued solutions of (9.1), though the presented approach is applicable in a complex-valued situation as well (instead of complex Vekua equations their bicomplex generalizations should be considered, see Section 14.3). We consider equation (9.1) on a plane of the variables x and y, that is α and $\overrightarrow B$ are functions of two Cartesian variables only. In this case, as we show in Section 9.2, equation (9.1) reduces to the equation

$$ div\left( {\frac{1} {\alpha }\nabla u} \right) + \alpha u = 0. $$

(9.2)

.