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

Temporally sparse data assimilation for the small-scale reconstruction of turbulence