Taking advantage of alternative expressions for potential vorticity (PV) in divergence forms, we derive balances between volume integral of PV and boundary conditions, that are then applied to practical computations of PV:

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we propose a new method for diagnosing the Ertel potential vorticity from model output, that preserves the balances;

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we show how the expression of PV can be derived in general coordinate systems. This is here emphasised with isopycnic coordinates by generalising the PV expression to the general Navier-Stokes equations;

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we propose a generalised derivation for the Haynes-McIntyre impermeability theorem, which highlights the role of the bottom boundary condition choice (e.g. no-slip vs free-slip) and mixing near the bottom boundary for the volume integral of PV.

The implications of balances between volume integral of PV and boundary conditions are then analysed for specific processes at various scales:

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at large scale, we show how an integral involving surface observations (derived from satellite observations) is linked to the integral of PV within a layer (between two isopycnals). This surface integral can be calculated for models and observations and can be used for validation;

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at mesoscale or sub-mesoscale, we analyse the relationship between net PV anomalies and net surface density anomalies for idealised vortices and 2D fronts. This can help determining vortex or jet structures for idealised studies or empirical methodologies;

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we also confirm and integrate previous results on the modification of PV within a bottom boundary layer into a single diagnostic taking into account the effect of density and velocity modifications by diabatic processes along the topography and diapycnal mixing within the boundary layer.