Near-cancellation of enstrophy and strain requires shear or a balanced vortex

Abstract

In the nonlinear source of the pressure Poisson equation the fluctuating enstrophy and strain nearly cancel. We ask where the cancellation is complete, which we call silent, and how the source escapes it. Pointwise the source depends only on the eigenvalues of the velocity gradient. Pure shear has none, so it is silent. It is the origin of the normalised invariant plane, whose second-invariant coordinate is the normalised source. Let $f$ be the shear fraction of the gradient and $m$ its distance from that origin. We prove that $m\le1-f\le3m$ at every point of every incompressible flow. Wherever the source nearly vanishes and the third invariant is small, the gradient is nearly pure shear. Otherwise the cancelling gradient is a balanced vortex, a swirl its own strain cancels at any strength. At a no-slip wall the silent object is exact: the wall shear-stress fluctuation, a rank-one sheet. Away from the wall we assume the same sheet, dressed with a small residual, and derive how it escapes. The escape passes through one element of the residual, the along-sheet variation of the sheet-normal velocity. Its amplitude is the geometric mean of sheet and residual, and its sign selects swirl or strain. A rank-two silent object has the same source through the same gates but balanced eigenvalues, so the assumption fixes the escape's topology, not its source. The shear fraction is thus a pointwise proxy for cancellation by shear, and a model of the source must be built on what escapes.