Mars Global Surveyor measurements revealed that the Martian crust is strongly magnetized in the southern hemisphere while the northern hemisphere is virtually void of magnetization. Two possible reasons have been suggested for this dichotomy: a once more or less homogeneously magnetization may have been destroyed in the northern hemisphere by, for example, resurfacing or impacts. The alternative theory we further explore here assumes that the dynamo itself produced a hemispherical field (Stanley et al., 2008; Amit et al., 2011). We use numerical dynamo simulations to study under which conditions a spatial variation of the heat flux through the core-mantle boundary (CMB) may yield a strongly hemispherical surface field. We assume that the early Martian dynamo was exclusively driven by secular cooling and we mostly concentrate on a cosine CMB heat flux pattern with a minimum at the north pole, possibly caused by the impacts responsible for the northern lowlands. This pattern consistently triggers a convective mode which is dominated by equatorially anti-symmetric and axisymmetric (EAA, Landeau and Aubert, 2011) thermal winds. Convective up- and down-wellings and thus radial magnetic field production then tend to concentrate in the southern hemisphere which is still cooled efficiently while the northern hemisphere remains hot. The dynamo changes from an α2 for a homogeneous CMB heat flux to an αΩ-type in the hemispherical configuration. These dynamos reverse on time scales of about 10 kyrs. This too fast to allow for the more or less unidirectional magnetization of thick crustal layer required to explain the strong magnetization in the southern hemisphere.