  We study the use of the Azouani-Olson-Titi (AOT) continuous data assimilation
algorithm to recover solutions of the Navier--Stokes equations modified to have
higher-order fractional diffusion. The fractional diffusion case is of
particular interest, as it is known to be globally well-posed for sufficiently
large diffusion exponent $\alpha$. In this work, we prove that the assimilation
equations are globally well-posed, and we demonstrate that the solutions
produced by the AOT algorithm exhibit exponential convergence in time to the
reference solution, given a sufficiently high spatial resolution of
observations and a sufficiently large nudging parameter.
  We also note that the results hold in spatial dimensions $d$ where $2\leq
d\leq 8$, so long as $\alpha\geq \frac12 +\frac{d}{4}$. Though the cases
$3<d\leq8$ are likely only a mathematical curiosity, we include them as they
cause no additional difficulty in the proof. Note that we show in a companion
paper the $d=2$ case allows for $\alpha<1$.
