Formula omitted]-Coordinate, an Arbitrary Lagrangian–Eulerian coordinate separating high and low frequency motions

Abstract

A new geopotential based vertical coordinate, named z˜, is presented, that treats external motions as a z∗-coordinate, internal low frequency motions in a Eulerian way and high frequency ones in a Lagrangian way. To that purpose, a prognostic equation for the low frequency horizontal flux divergence is added in the system in order to discriminate between low and high frequency parts of vertical motions. This coordinate keeps the intrinsic advantages of the z-coordinate while enabling a drastic reduction of the spurious diapycnal mixing induced by vertical and lateral schemes truncation errors associated with high frequency motions in level models. An idealised experiment illustrates this property in an internal wave propagation framework. The reduction of Eulerian vertical velocities enables a proportional reduction of the associated advection schemes truncation errors. Following high frequency motions, the z˜-coordinate lets lateral schemes operate along tracer anomalies created by those motions.