Simultaneous Multi-Wavelength Observations of Magnetic Activity in Ultracool Dwarfs. I. The Complex Behavior of the M8.5 Dwarf TVLM513-46546

[Abridged] We present the first simultaneous radio, X-ray, ultraviolet, and optical spectroscopic observations of the M8.5 dwarf TVLM513-46546, with a duration of 9 hours. These observations are part of a program to study the origin of magnetic activity in ultracool dwarfs, and its impact on chromospheric and coronal emission. Here we detect steady quiescent radio emission superposed with multiple short-duration, highly polarized flares; there is no evidence for periodic bursts previously reported for this object, indicating their transient nature. We also detect soft X-ray emission, with L_X/L_bol~10^-4.9, the faintest to date for any object later than M5, and a possible weak X-ray flare. TVLM513-46546 continues the trend of severe violation of the radio/X-ray correlation in ultracool dwarfs, by nearly 4 orders of magnitude. From the optical spectroscopy we find that the Balmer line luminosity exceeds the X-ray luminosity by a factor of a few, suggesting that, unlike in early M dwarfs, chromospheric heating may not be due to coronal X-ray emission. More importantly, we detect a sinusoidal H-alpha light curve with a period of 2 hr, matching the rotation period of TVLM513-46546. This is the first known example of such Balmer line behavior, which points to a co-rotating chromospheric hot spot or an extended magnetic structure, with a covering fraction of about 50%. This feature may be transitory based on the apparent decline in light curve peak during the four observed maxima. From the radio data we infer a large scale steady magnetic field of ~100 G, in good agreement with the value required for confinement of the X-ray emitting plasma. The radio flares, on the other hand, are produced in a component of the field with a strength of ~3 kG and a likely multi-polar configuration.