We study the new mechanism of the axion production suggested recently in [1,2]. This mechanism is based on the so-called Axion Quark Nugget (AQN) dark matter model, which was originally invented to explain the similarity of the dark and visible cosmological matter densities. We perform numerical simulations to evaluate the axion flux on the Earths surface. We examine annual and daily modulations, which have been studied previously and are known to occur for any type of dark matter. We also discuss a novel type of short time enhancements which are unique to the AQN model: the statistical fluctuations and burst-like amplification, both of which can drastically amplify the axion signal, up to a factor $sim10^2-10^3$ for a very short period of time. The present work studies the AQN-induced axions within the mass window $10^{-6}{rm,eV}lesssim m_alesssim10^{-3}rm,eV$ with typical velocities $langle v_aranglesim0.6c$. We also comment on the broadband detection strategy to search for such relativistic axions by studying the daily and annual time modulations as well as random burst-like amplifications.
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