  The evolution of a determining form for the 2D Navier-Stokes equations (NSE),
which is an ODE on a space of trajectories is completely described. It is
proved that at every stage of its evolution, the solution is a convex
combination of the initial trajectory and the fixed steady state, with a
dynamical convexity parameter $\theta$, which will be called the characteristic
determining parameter. That is, we show a remarkable separation of variables
formula for the solution of the determining form. Moreover, for a given initial
trajectory, the dynamics of the infinite-dimensional determining form are
equivalent to those of the characteristic determining parameter $\theta$ which
is governed by a one-dimensional ODE. %for the parameter specifying the
position on the line segment. This one-dimensional ODE is used to show that if
the solution to the determining form converges to the fixed state it does so no
faster than $\mathcal{O}(\tau^{-1/2})$, otherwise it converges to a projection
of some other trajectory in the global attractor of the NSE, but no faster than
$\mathcal{O}(\tau^{-1})$, as $\tau \to \infty$, where $\tau$ is the
evolutionary variable in determining form. The one-dimensional ODE also
exploited in computations which suggest that the one-sided convergence rate
estimates are in fact achieved. The ODE is then modified to accelerate the
convergence to an exponential rate. Remarkably, it is shown that the zeros of
the scalar function that governs the dynamics of $\theta$, which are called
characteristic determining values, identify in a unique fashion the
trajectories in the global attractor of the 2D NSE. Furthermore, the
one-dimensional characteristic determining form enables us to find
unanticipated geometric features of the global attractor, a subject of future
research.
