Streamfunction advection bracket ($J(f,g)=f_z{g}_x-f_x{g}_z$)

For the two-dimensional streamfunction convention $u={\psi }_z$, $w=-{\psi }_x$, the advective derivative of $\theta$ is $u{\theta }_x+w{\theta }_z=J(\psi ,\theta )$. This antisymmetric Jacobian determinant bracket organizes nonlinear interactions in convection and vorticity equations.