Vorticity in a simulation of a two-dimensional incompressible fluid. The panels show, respectively: instantaneous vorticity $\zeta$, GLM mean vorticity $\bar{\zeta}$, volume-preserving mean vorticity $\bar{\zeta}_{\dagger}$, and the mean obtained using the divergence-free projection of the GLM mean velocity. An exponential average with $\alpha = 0.5$ is used.
Vorticity field of the shallow-water simulation: the instantaneous vorticity $\zeta$ is compared with its bare and volume-preserving exponential Lagrangian averages $\bar{\zeta}$ and $\bar{\zeta}_{\dagger}$.