Phase transport defect

For a velocity field $u$ and a smooth phase $\Phi$ , the phase transport defect is the material derivative

E_\Phi=(\partial_t+u\cdot\nabla)\Phi.

It vanishes when $\Phi$ is transported exactly along the particle trajectories of $u$ . The velocity and the derivatives held fixed are part of this definition.

Effect on an oscillatory field

Writing $D_t=\partial_t+u\cdot\nabla$ ,

D_t(ae^{i\kappa\Phi})=e^{i\kappa\Phi}(D_ta+i\kappa E_\Phi a).

Even a small phase defect is multiplied by the carrier frequency. Moreover, for $n=\nabla\Phi$ ,

D_tn=-(\nabla u)^Tn+\nabla E_\Phi,

where $(\nabla u)_{ij}=\partial_j u_i$ . Exact transport therefore evolves the phase gradient by the transpose velocity gradient.