That horizontal wind velocity for which the Coriolis acceleration exactly balances the horizontal pressure force:

where vg is the geostrophic wind, f the Coriolis parameter, k the vertical unit vector, g the acceleration of gravity, ∇p the horizontal del operator with pressure as the vertical coordinate, and z the height of the constant-pressure surface. The geostrophic wind is thus directed along the contour lines on a constant-pressure surface (or along the isobars in a geopotential surface) with low elevations (or low pressure) to the left in the Northern Hemisphere and to the right in the Southern Hemisphere. The speed of the geostrophic wind Vg is given by

where ∂z/∂n is the slope of the isobaric surface normal to the contour lines to the left of the direction of motion in the Northern Hemisphere and to the right in the Southern Hemisphere. The geostrophic wind is defined at every point except along the equator. The validity of this approximation in dynamic theory depends upon the particular context. See geostrophic equilibrium, geostrophic approximation, quasigeostrophic approximation; compare gradient wind.