Ideal magnetohydrodynamics (MHD) still provides the mathematical framework and the textbook vocabulary in which the possible states of a toroidal plasma are discussed, generally regarded as static equilibria. This is so, despite the increasing realization that virtually all toroidal magnetofluids have nontrivial fluid flows (finite velocity fields) in them. A very different perspective results from nonideal MHD, including both resistivity and viscosity and invoking nonideal boundary conditions. There, it has been shown that if Ohm's law and Faraday's law are given equal importance with force balance, flows are an inevitable consequence of the assumptions of time independence and axisymmetry. Previous treatments of the toroidal steady states for such systems have been based on perturbation theory in which the flow velocity was assumed small, as a consequence of high viscosity (or in dimensionless terms, low Hartmann number H). Here, recently newly available numerical programs are used to lift this limitation and to solve nonlinearly for the allowed steady states of an axisymmetric, current-carrying, toroidal magnetofluid without such an expansion in the Hartmann number. Flow patterns for values of H from 1 to 1 have been calculated. As H is raised, the flow pattern goes from the predominantly poloidal pair of counter-rotating "convection cells" revealed by the perturbation theory to a pattern in which the toroidal kinetic energy of flow considerably exceeds the poloidal kinetic energy. In no case is the flow discovered a simple rotation.