Dynamo action in cellular convection

The dynamo properties of square patterns in the Rayleigh-Bénard Boussinesq convection in a plane horizontal layer are studied numerically. Cases without rotation and with a weak rotation about a vertical axis are considered, particular attention being paid to the relation between the dynamo action and the kinetic helicity of the flow. While the fluid layer is symmetric with respect to up-down reflections, the square-pattern solutions may or may not possess this vertical symmetry. Vertically symmetric solutions, appearing in the form of checkerboard patterns, do not possess a net kinetic helicity and we find them to be incapable of dynamo action at least up to magnetic Reynolds numbers of ≈ 12000. Vertically asymmetric squares, a secondary convection pattern appearing via the skewed varicose instability of rolls and being characterized by a rising (descending) motion in the centers and a descending (rising) motion near the boundaries, can in turn be devided into such that possess a full horizontal square symmetry and others lacking this symmetry. The flows lacking both the vertical and horizontal symmetries are particularly interesting in that they possess kinetic helicity and show kinematic dynamo action even without rotation. The generated magnetic fields are concentrated in vertically oriented filamentary structures near cell boundaries. The dynamos found in the nonrotating case are, however, always only kinematic, never nonlinear dynamos. Nonlinearly the back-reaction of the magnetic field then forces the flow into the basin of attraction of a roll-pattern solution incapable of dynamo action. But with rotation added, parameter regions are found, where a subtle balance between the Coriolis and Lorentz forces enables a nonlinear dynamo action of stationary asymmetric squares. In some parameter regions this balance leads to nonlinear dynamos with flows in the form of oscillating squares or stationary modulated rolls