Cascades from $\nu_e$ above $10^{20}$ eV

At very high energies, the Landau-Pomeranchuk-Migdal effect reduces the cross sections for electron bremsstrahlung and photon $e^+e^-$ pair production. The fractional electron energy loss and pair production cross sections drop as the energy increases. In contrast, the cross sections for photonuclear interactions grow with energy. In solids and liquids, at energies above $10^{20}$ eV, photonuclear reactions dominate, and showers that originate as photons or electrons quickly become hadronic showers. These electron-initiated hadronic showers are much shorter (due to the absence of the LPM effect), but wider than purely electromagnetic showers would be. This change in shape alters the spectrum of the electromagnetic and acoustic radiation emitted from the shower. These alterations have important implications for existing and planned searches for radiation from $\nu_e$ induced showers above $10^{20}$ eV, and some existing limits should be reevaluated.