Resonant mode of a directional derivative

For $D_v=v\cdot\nabla$ on the unit torus, a Fourier mode $m\in\mathbb Z^n$ is resonant if $v\cdot m=0$ , since

D_v e^{2\pi i m\cdot x}=2\pi i(v\cdot m)e^{2\pi i m\cdot x}.

The zero mode is always resonant. A nonzero resonant mode is an additional obstruction to solving $D_vu=f$ : the corresponding Fourier coefficient of $f$ must vanish.

Near resonance

A nearly resonant mode has a small nonzero eigenvalue relative to a specified tolerance. It need not lie in the kernel.