Highest Energy Neutrino Showers in EUSO

EUSO experiment, while monitoring the downward Earth atmosphere layers, may observe among common Ultra High Energy Cosmic Rays, UHECR, also High Energy Neutrino-Induced Showers either blazing upward to the detectors at high (PeVs) energies or at much higher GZK E >= 10^19 eV energies, showering horizontally in air or vertically downward. A small fraction of these upward, horizontal and vertical Shower maybe originated by direct astrophysical UHE neutrino interacting on terrestrial air layers itself; however the dominant UHE neutrino signal are Upward and Horizontal Tau Air-Showers, UPTAUS and HORTAUs (or Earth skimming neutrinos), born within widest Earth Crust Crown (Sea or Rock) Areas, by UHE nu_tau + Nuclei--> tau interactions, respectively at PeVs and GZK energies: their rate and signatures are shown in a neutrino fluence map for EUSO thresholds versus other UHE air interacting neutrino signals and backgrounds. The effective target Masses originating HORTAUs seen by EUSO may exceed (on sea) at 10^19 eV energy a wide and huge ring volume above 1/2*A_Euso*l_tau*(L_nu/2R_Earth)^2= 5130 km^3. The consequent HORTAUS event rate (even at 10% EUSO duty cycle lifetime) may deeply test the expected Z-Burst models by at least a hundred of yearly events. Even rarest but inescapable GZK neutrinos (secondary of photopion production of observed cosmic UHECR) might be discovered in a few (or a tens) horizontal shower events; in this view an extension of EUSO detectability up to E > 10^19 eV threshold is to be preferred. A wider collecting EUSO telescope (3m diameter) might be considered.