  Analyzing the fluid velocity gradients in a Lagrangian reference frame
provides an insightful way to study the small-scale dynamics of turbulent
flows, and further insight is provided by considering the equations in the
eigenframe of the strain-rate tensor. The dynamics of the velocity gradient
tensor is governed in part by the anisotropic pressure Hessian, which is a
non-local functional of the velocity gradient field. This anisotropic pressure
Hessian plays a key role in the velocity gradient dynamics, for example in
preventing finite-time singularities, but it is difficult to understand and
model due to its non-locality and complexity. In this work a gauge symmetry for
the pressure Hessian is introduced to the eigenframe equations of the velocity
gradient, such that when the gauge is added to the original pressure Hessian,
the dynamics of the eigenframe variables remain unchanged. We then exploit this
gauge symmetry to perform a rank reduction on the three-dimensional anisotropic
pressure Hessian, which, remarkably, is possible everywhere in the flow. The
dynamical activity of the newly introduced rank-reduced anisotropic pressure
Hessian is confined to two dimensional manifolds in the three dimensional flow,
and exhibits striking alignment properties with respect to the strain-rate
eigenframe and the vorticity vector. The dimensionality reduction, together
with the strong preferential alignment properties, leads to new dynamical
insights for understanding and modelling the role of the anisotropic pressure
Hessian in three-dimensional flows.
