The winter polar ionosphere, under southward interplanetary magnetic field (IMF) conditions, experiences irregularity development leading to consequences such as scintillation on transionospheric communication links. These irregularities are associated with antisunward convecting polar ionospheric patches. The gradient drift instability (GDI) has been considered a primary candidate for the generation of these irregularities, or at least the long-wavelength energy source of the irregularity-wave cascade process. The Utah State University time-dependent ionospheric model (TDIM) enables the polar cap ionosphere and its patches to be modeled on a large scale in a time-evolving manner. Hence, at each point in space and time, the TDIM has adequate information to compute the growth rate of the gradient drift instability. The spatial gradients are based on a 92 × 92 km TDIM grid resolution, while the time resolution can be as short as 30 s. In this study, we present results of the GDI linear growth rate calculations for the F region. For a first time, snapshots of the instantaneous GDI linear growth rates over the polar ionosphere are shown. These show, in part, the correlation of the GDI with polar cap patches. Then we present the time evolution of the linear growth rate as plasma flux tubes convect. It is shown that a flux tube of plasma can, in fact, change between being unstable and stable to the GDI as the plasma convection pattern switches in response to changing IMF conditions. These convective results are essential information if an assessment of whether or not a plasma flux tube is stable to the gradient drift mechanism is to be made.