The gamma plane approximation introduced in this study corresponds to a nonlinear horizontal shallow flow in a plane where, in addition to the familiar linear variation off(i.e., β), there is a quadratic variation with latitude. Such a plane may have some application to the mesoscale oceanic flow in the immediate vicinity of the North Pole because at the pole the linear gradient (β) vanishes so that the quadratic variation (γ) is the dominant gradient. It is also applicable to the flow near the center of a rotating (laboratory) tank.