On the interface equation for the inversion of balance equations

Balance equations (BEs) which describe the balance constraints among balanced variables have wide applications in data assimilation and diagnosing balanced motions. In the applications an additional equation serving as an interface to connect the balanced variables and full elds (data assimilation analyses or elds of primitive equations) is required for the inversion of BEs to obtain the balanced variables. The interface equation is often constructed through the potential vorticity (PV) based on the argument that PV of the full elds should be preserved in the balanced motion. However, it is not clear that for the given balance constraints this prescribed PV‐based interface equation can lead to the optimal solutions for the BEs. To search for the interface equation which can produce the optimal solutions for the BEs under given balance constraints, the variational approach is proposed to derive instead of ‘prescribing’ the interface equation by minimizing an energy‐like norm under both geostrophic and Charney balance constraints. The derived interface equations are compared against the PV‐based interface equation for the two‐dimensional (2D) shallow‐water model and three‐dimensional (3D) baroclinic system. Analytical results show that the two kinds of interface equation are dierent in general but are close when the Rossby number and the departure of geopotential/temperature from its reference value are small. The accuracy associated with the derived interface equation is examined by comparing the balanced variables obtained by inverting the BEs with the derived and PV‐based interface equations numerically in a hemisphere shallow‐water system. The results show that under the same balance constraint the derived interface equation leads to better global accuracy of balanced variables.