  The Lagrangian dynamics of the velocity gradient tensor A in isotropic and
homogeneous turbulence depend on the joint action of the self-streching term
and the pressure Hessian. Existing closures for pressure effects in terms of A
are unable to reproduce one important statistical role played by the
anisotropic part of the pressure Hessian, namely the redistribution of the
probabilities towards enstrophy production dominated regions. As a step towards
elucidating the required properties of closures, we study several synthetic
velocity fields and how well they reproduce anisotropic pressure effects. It is
found that synthetic (i) Gaussian, (ii) Multifractal and (iii) Minimal Turnover
Lagrangian Map (MTLM) incompressible velocity fields reproduce many features of
real pressure fields that are obtained from numerical simulations of the Navier
Stokes equations, including the redistribution towards enstrophy-production
regions. The synthetic fields include both spatially local, and nonlocal,
anisotropic pressure effects. However, we show that the local effects appear to
be the most important ones: by assuming that the pressure Hessian is local in
space, an expression in terms of the Hessian of the second invariant Q of the
velocity gradient tensor can be obtained. This term is found to be well
correlated with the true pressure Hessian both in terms of eigenvalue
magnitudes and eigenvector alignments.
