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The Askaryan Radio Array (ARA) is an ultra-high energy (UHE, $>10^{17}$ eV)neutrino detector designed to observe neutrinos by searching for the radiowaves emitted by the relativistic products of neutrino-nucleon interactions inAntarctic ice. In this paper, we present constraints on the diffuse flux ofultra-high energy neutrinos between $10^{16}-10^{21}$ eV resulting from asearch for neutrinos in two complementary analyses, both analyzing four yearsof data (2013-2016) from the two deep stations (A2, A3) operating at that time.We place a 90 % CL upper limit on the diffuse all flavor neutrino flux at$10^{18}$ eV of $EF(E)=5.6\times10^{-16}$$\textrm{cm}^{-2}$$\textrm{s}^{-1}$$\textrm{sr}^{-1}$. This analysis includesfour times the exposure of the previous ARA result, and representsapproximately 1/5 the exposure expected from operating ARA until the end of2022.

Constraints on the Diffuse Flux of Ultra-High Energy Neutrinos from Four Years of Askaryan Radio Array Data in Two Stations