A new derivation and interpretation of the semigeostrophic (SG) material invariant in the theory of geophysical flows is introduced. First, a generalized three-dimensional equation of the SG dynamics is established and the generalized equations for the rate of change of vorticity and for the rate of change of the velocity gradient cofactor tensor are obtained. Next, a conservation equation for the vorticity-velocity gradient cofactor tensor (denoted Xi) is derived. The specific potential Xi, that is, Xi in the reference configuration per unit of mass, is defined and an expression for its rate of change is obtained. The SG invariant is interpreted as the vertical component of the specific potential Xi. Under the SG assumptions (advection of the geostrophic velocity, hydrostatic, and f-plane approximations) this vertical component is materially conserved in the SG flow. The generalized SG invariant (i.e., the specific potential Xi) differs conceptually from the Beltrami-Rossby-Ertel specific potential vorticity. Its conservation in the SG flow seems to be highly dependent on the SG assumptions, especially on the f-plane approximation and on the horizontal nature of the geostrophic velocity.
