On the generalized Ertel-Rossby invariant

A new invariant called the generalized Ertel-Rossby invariant (GER) was developed in this study. The new invariant is given by the dot product of the generalized vorticity and the generalized velocity. The generalized vorticity is the absolute vorticity minus the cross-product of the gradient of Lagrangian-time integrated temperature and the gradient of entropy. The generalized velocity is the absolute velocity minus the sum of the gradient of Lagrangian-time integrated kinetic potential and the Lagrangian-time integrated temperature multiplied by the gradient of entropy. In addition to the traditional potential vorticity, the GER invariant may provide another useful tool to study the atmospheric dynamic processes for weather phenomena ranging from large scales to small scales.