The surface of Mars appears dramatically different between the northern and southern hemispheres. Any endogenic origin for this hemispheric dichotomy must involve a pattern of mantle convection that reflects the shape of the dichotomy, primarily spherical harmonic degree-1. We investigated two mechanisms by which degree-1 convection may be initiated in the Martian mantle: (1) an endothermic phase change near the CMB and (2) viscosity layering in the mid-mantle. Using two-dimensional (2-D) and 3-D spherical finite-element convection models, we explored the conditions under which each mechanism can produce degree-1 structures. The phase transition is only effective at generating degree-1 structures when the mantle viscosity is constant or weakly temperature-dependent (activation energy <100 kJ/mol), but the degree-1 pattern requires several billion years to develop. Increasing convective vigor in phase change models leads to reduced wavelengths for convective structures. Degree-1 convection can also develop in a layered viscosity mantle, with temperature- and depth-dependent viscosity. An overall sublithospheric radial viscosity variation of a factor of 100 including a factor of 8--25 jump in the midmantle can lead to formation of degree-1 structure in a timescale ranging from 100 My to several hundred My, consistent with the timescale for the formation of the dichotomy. Neither convective vigor nor the internal heating rate greatly affects the formation of degree-1 structures. We propose that degree-1 mantle convection induced by a layered viscosity structure may be responsible for the formation of the crustal dichotomy.