A thermobaric instability of Lagrangian vertical coordinate ocean models

Lagrangian- (and isopycnic-) vertical coordinate ocean models are subject to an exponentially growing numerical instability in weakly stratified regions when thermobaricity is not accurately compensated. Inaccurate compensation for compressibility in the pressure gradient terms leads to pressure gradient truncation errors (due to the vertical discretization) that can drive the Lagrangian coordinate surfaces to reinforce these errors. It is possible to avoid this instability while using the full non-linear equation of state for seawater by using an optimal alternate discretization of the pressure gradient terms and extracting a slowly spatially varying reference compressibility that approximates the compressibility of the ocean's mean state.