This dissertation concerns the hydrography of hydrothermal plumes and the problem of measuring the flux of heat from submarine volcanoes to the deep sea. It presents the results of an experiment designed to measure the heat flux from the Main Endeavour hydrotherml vent field (MEF) on the Endeavour segment of the Juan de Fuca ridge in the deep Northeast Pacific. Primary data are from an autonomous underwater vehicle called Autonomous Benthic Explorer (ABE), a lowered CTD, and 2 current meter moorings. Chapter~\ref{ch:power} motivates heat flux measurement with questions about the nature of the subsurface oceanic crust and the influence of hydrothermal venting on the oceans. Context for the Flow Mow experiment is then established through definition of terminology, a review of historical heat flux measurements, and a re-examination of plume theory. Chapter~\ref{ch:flowmow} presents the field program and experimental setting, emphasizing currents and hydrography near the MEF. A key observation is that axial currents change dramatically near the depth of the ridge crests --- from rectified, oscillatory flow within the axial valley to relatively fast, rotary flow in the few hundred meters above the ridge crests. The remainder of the thesis is devoted to the problem of estimating flux in the distinct environments: the axial valley (Chapter~\ref{ch:lower}) and above the ridge (Chapter~\ref{ch:upper}). In both parts, a ``puff'' model of plume advection and diffusion is utilized to understand the field observations, as well as to predict the patterns of variability that can be generated by combinations of oscillatory and mean flow.