The situation for the baroclinic circulation is more complicated, but a similar tool can be introduced.

For a steady flow with weak friction in the ocean interior, regardless of whether there are cross-isopycnal fluxes, potential vorticity conservation is reduced to

where un is the horizontal velocity in the n-th layer (Pedlosky, 1996). Equation (4.452) can be seen as a direct result of geostrophic balance on isopycnal surfaces, and allows us to introduce a virtual streamfunction fn for flow on the n-th isopycnal layer

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