The equations of magnetohydrodynamics (MHD) describe the interaction of an electrically conductive fluid with an ambient magnetic field. In this talk we consider the nonlinear interaction of a shear flow and constant magnetic field in the two-dimensional viscous, non-resistive magnetohydrodynamics equations (MHD). The anisotropic dissipation breaks symmetry and leads to novel instabilities of the induced current due to dynamo effects. This is based on joint work with Michele Dolce and Niklas Knobel.