Shearing-periodic boundary condition ($f(L_x,y,z,t)=f(0,y+2A{L}_{x}t,z,t)$)

For background velocity $-2Ax{\mathbf{e}}_y$, opposite radial faces of a shearing box are identified after the displayed azimuthal shift, modulo $L_y$. Products and spatial derivatives inherit this condition. Their radial surface integrals cancel because a translation preserves an integral over a periodic azimuthal interval.

For zero-mean fields in a shearing box, Fourier modes have wavevectors $(2\pi n/L_x+2At2\pi m/L_y,2\pi m/L_y,2\pi p/L_z)$. A nonzero azimuthal or vertical index supplies at least $2\pi /L$, where $L$ is the largest box side; otherwise the nonzero radial index does. Parseval identity gives the displayed Poincaré inequality uniformly in time.

Periodic azimuthal and vertical boundary terms vanish. Radial boundary terms cancel after the azimuthal translation prescribed by the shearing-periodic boundary condition. Consequently the volume average of a divergence is zero, enabling integration by parts for the smooth shearing-periodic fields.