Wave activity ($\mathcal{A}$)

Wave activity is a conserved quadratic measure of a small disturbance when the basic state has a continuous symmetry. Its flux divergence gives the exchange of pseudomomentum between waves and the mean flow.

For a small-amplitude conservative wave packet in a slowly varying fluid flow, its intrinsic energy divided by intrinsic frequency is transported along its group velocity, with the basic advection added in the laboratory frame. A stationary background conserves laboratory angular frequency, but a changing mean velocity changes the intrinsic one through the Doppler effect; intrinsic energy alone need not be conserved. Geometrical spreading and dissipation enter the action-flux balance.

The wavevector times wave action gives the canonical momentum associated with translations of a wave's phase. It describes momentum exchange with a slowly varying mean fluid flow, rather than simply the spatial average of oscillating fluid momentum. For horizontal wavenumber $k$, its horizontal component is $kE/\hat{\omega }$. A convention using displacement pseudomomentum of an internal gravity wave instead has the opposite sign.

For small quasi-geostrophic approximation waves on a locally uniform basic zonal flow with nonzero meridional potential-vorticity gradient $Q_y$, this signed quadratic wave activity obeys ${\mathcal{A}}_t+\nabla \cdot \mathbf{F}=\overline{q^{\prime }\mathcal{D}}/Q_y$, where $\mathbf{F}$ is the Eliassen–Palm flux and $\mathcal{D}$ is a PV source. Multiply the linear PV equation by $q^{\prime }/Q_y$ and use the Taylor identity for quasi-geostrophic flux to obtain it. In a locally uniform conservative wave packet, $\mathbf{F}=\mathcal{A}{\mathbf{c}}_g$, so the flux follows the group velocity, with its sign determined by $Q_y$.