Using data obtained by the high resolution CRisp Imaging SpectroPolarimeter instrument on the Swedish 1-m Solar Telescope, we investigate the dynamics and stability of quiet-Sun chromospheric jets observed at disk center. Small-scale features, such as Rapid Redshifted and Blueshifted Excursions, appearing as high speed jets in the wings of the H$alpha$ line, are characterized by short lifetimes and rapid fading without any descending behavior. To study the theoretical aspects of their stability without considering their formation mechanism, we model chromospheric jets as twisted magnetic flux tubes moving along their axis, and use the ideal linear incompressible magnetohydrodynamic approximation to derive the governing dispersion equation. Analytical solutions of the dispersion equation indicate that this type of jet is unstable to Kelvin-Helmholtz instability (KHI), with a very short (few seconds) instability growth time at high upflow speeds. The generated vortices and unresolved turbulent flows associated with the KHI could be observed as broadening of chromospheric spectral lines. Analysis of the H$alpha$ line profiles shows that the detected structures have enhanced line widths with respect to the background. We also investigate the stability of a larger scale H$alpha$ jet that was ejected along the line-of-sight. Vortex-like features, rapidly developing around the jets boundary, are considered as evidence of the KHI. The analysis of the energy equation in the partially ionized plasma shows that the ion-neutral collisions may lead to the fast heating of the KH vortices over timescales comparable to the lifetime of chromospheric jets.
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