We present time-series, high-resolution optical spectroscopy of the eccentric T Tauri binary TWA 3A. Our analysis focuses on variability in the strength and structure of the accretion tracing emission lines H alpha and He I 5876A. We find emission line strengths to display the same orbital-phase dependent behavior found with time-series photometry, namely, bursts of accretion near periastron passages. Such bursts are in good agreement with numerical simulations of young eccentric binaries. During accretion bursts, the emission of He I 5876A consistently traces the velocity of the primary star. After removing a model for the systems chromospheric emission, we find the primary star typically emits ~70% of the He I accretion flux. We interpret this result as evidence for circumbinary accretion streams that preferentially feed the TWA 3A primary. This finding is in contrast to most numerical simulations, which predict the secondary should be the dominant accretor in a binary system. Our results may be consistent with a model in which the precession of an eccentric circumbinary disk gap alternates between preferentially supplying mass to the primary and secondary.
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