  Previous works have shown that the small-scale information of incompressible
homogeneous isotropic turbulence (HIT) is fully recoverable as long as
sufficient large-scale structures are continuously enforced through temporally
continuous data assimilation (TCDA). In the current work, we show that the
assimilation time step can be relaxed to values about 1 $\sim$ 2 orders larger
than that for TCDA, using a temporally sparse data assimilation (TSDA)
strategy, while the accuracy is still maintained or even slightly better in the
presence of non-negligible large-scale errors. The one-step data assimilation
(ODA) is examined to unravel the mechanism of TSDA. It is shown that the
relaxation effect for errors above the assimilation wavenumber $k_a$ is
responsible for the error decay in ODA. Meanwhile, The errors contained in the
large scales can propagate into small scales and make the high-wavenumber
($k>k_a$) error noise decay slower with TCDA than TSDA. This mechanism is
further confirmed by incorporating different levels of errors in the large
scales of the reference flow field. The advantage of TSDA is found to grow with
the magnitude of the incorporated errors. Thus, it is potentially more
beneficial to adopt TSDA if the reference data contains non-negligible errors.
Finally, an outstanding issue raised in previous works regarding the
possibility of recovering the dynamics of sub-Kolmogorov scales using direct
numerical simulation (DNS) data at Kolmogorov scale resolution is also
discussed.
