A Generalization of the Sawyer-Eliassen Equation for a 3-D Domain

If we wish to determine the ageostrophic circulation of a large-scale 3-D disturbance in a balanced flow, we would need to generalize the Sawyer-Eliassen equation which is well known for governing the ageostrophic circulation in a large-scale 2-D disturbance. We will present such generalization for a 3-D semi-geostrophic model. A formal analytic solution of the generalized equations will be presented. In a trial application of this solution in conjunction with the geostrophic streamfunction of an Eady wave, the field of the horizontal components of the ageostrophic velocity at any vertical level is found to have the appearance of a magnetic field. If time permits, an application of it in the context of an investigation of 3-D frontal dynamics will also be discussed.