\section{Introduction and motivation}

The heat transferred from oceanic crust to sea water during hydrothermal 
circulation is 

\begin{equation}
H_{rock} = {\rho V c_p \Delta T}_{rock} 
\end{equation}

or 

\begin{equation}
H_{water} = \int_{\theta_{in}}^{\theta_{out}} m c_p d\theta
\end{equation}

Since it is unclear where hydrothermal fluid enters the crust, we assume that $\theta_{in} = \theta_B$, 
the temperature of background sea water at the depth of the vent emitting fluid at $\theta_{out}$.
Similarly, we assume that $S_{in} = S_B$.  These assumptions allow us to calculate a 
conservative tracer, $q$, that has zero concentration in the background NE Pacific sea water.
Subsequently, we can relate the flux of $q$ness in hydrothermal plumes 
(or an estimate of potential temperature anomaly, $\Delta\theta$) 
to source temperature, $\theta_{out}$.  The flux of heat from the cooling crust through some 
surface $A$ can then be expressed as 

\begin{equation}
HF_{water} = \rho c_p \Delta\theta v A.
\end{equation}

The fluid velocity $v$ may be vertical (in rising plumes) or horizontal (in plumes 
advected by currents).