  We formalise the concept of near resonance for the rotating Navier-Stokes
equations, based on which we propose a novel way to approximate the original
PDE. The spatial domain is a three-dimensional flat torus of arbitrary aspect
ratios. We prove that the family of proposed PDEs are globally well-posed for
any rotation rate and initial datum of any size in any $H^s$ space with
$s\ge0$. Such approximations retain much more 3-mode interactions, thus more
accurate, than the conventional exact resonance approach. Our approach is free
from any limiting argument that requires physical parameters to tend to zero or
infinity, and is free from any small divisor argument (so estimates depend
smoothly on the torus' aspect ratios). The key estimate hinges on counting of
integer solutions of Diophantine inequalities rather than Diophantine
equations. Using a range of novel ideas, we handle rigorously and optimally
challenges arising from the non-trivial irrational functions in these
inequalities. The main results and ingredients of the proofs can form part of
the mathematical foundation of a non-asymptotic approach to nonlinear
oscillatory dynamics in real-world applications.
