%======================================================================
%                    F L U X E S / A D V E C T I V E . T E X 
%                    doc: Tue Nov 16 12:39:38 1999
%                    dlm: Thu Jun  8 00:02:07 2000
%                    (c) 1999 A.M. Thurnherr
%                    uE-Info: 90 56 NIL 0 0 72 3 2 4 ofnI
%======================================================================

To derive an estimate which does not depend on any of the assumptions
required for the height-of-rise method \rsec{HF-hor}, an alternative
approach based on the advective flux of the equilibrium plume
temperature anomalies $\Delta\theta_e$\ls{De} is used
\cite{EPSL/baker+massoth87}. The Gaussian shape of the density-averaged
plume observed in 1997 \rsec{P-1997} supports the assumption of
quasi-steadiness, so that the equilibrium temperature-anomaly flux can
be defined as \begin{equation}
	%
	Q_e\overline{\Delta\theta_e} = \int\Delta\theta_e U_n dE
	\label{eqn:tanomflux}
	%
\end{equation} where $Q_e$ is the volume flux carrying the spatially
averaged temperature anomaly $\overline{\Delta\theta_e}$ away from the
``source'' (which in this context is the region where the buoyant
plumes become neutrally buoyant). The integral is evaluated over a
suitably defined section $E$\ls{E} across which the entire equilibrium
plume is advected with velocity $U_n$\ls{U_n} (normal to $E$) away from
the source. For a plume trapped in and advected unidirectionally along
the rift valley any cross section down-stream from the source can be
chosen for $E$ (background fluid advected across the section has zero
temperature anomalies by definition and does therefore not contribute to
the total flux).

If the $T/S$ relationship of the fluid entrained into the buoyant
plumes is linear the equilibrium temperature-anomaly flux
$Q_e\overline{\Delta\theta_e}$ is related to the corresponding source
temperature-anomaly flux $Q_i\overline{\Delta\theta_i}$ by \rexpr{mcd}.
The stability ratio for the 1997 data $\Rr=2.36$ is taken from
\rsecNP{NESPI-R} and the source-fluid density-anomaly ratio
$\Ri\approx5.4$ is estimated from the effluent properties measured in
the \R hydrothermal vent field approximately one month after the \F
cruise \cite{misc/fouquet+98}. The source heat flux $H$ is calculated
using \begin{equation}
	%
	H=\rho_i c_p Q_i\overline{\Delta\theta_i},
	\label{eqn:hflux}
	%
\end{equation} where $\rho_i$\ls{rho_i} is the ``asymptotic''
source-fluid density, i.e.\ calculated with the linear equation of
state expanded at background temperature and salinity, and $c_p$ is the
corresponding low-temperature specific heat
\cite{EPSL/turner+campbell87}.

%----------------------------------------------------------------------
% BILINEAR METHOD
%----------------------------------------------------------------------

Using the mean slope of the spice vs.\ nephel regressions of the 1997
\BGT data set \degCpV{C=-0.019} \rsec{N-1997} to re-scale the
near-source Gaussian light-scattering plume shown in
\rfigNP{F-bulkplumes} yields the temperature-anomaly profile
 %
\begin{equation}
	\Delta\theta_e(z) = C \times \Delta n^* 
		\times \exp\left( - \frac{(z-\mu)^2}{2\sigma^2} \right)
\end{equation}
 %
with \V{\Delta n^*\!=0.25}, \m{\mu=2120}, and \m{\sigma=96} taken from
the SW basin profile. The peak temperature anomaly becomes
\degC*{-4.8}{-3}, an order of magnitude less than the anomalies
predicted by the bulk plume model \rsec{HF-hor}.

%----------------------------------------------------------------------
% HEAT FLUX
%----------------------------------------------------------------------

The heat flux is now estimated using \rexprs{tanomflux}{hflux}. The
plume width is taken to be \km{2} \rsec{P-1997} and the mean
near-source velocity profile \rfig{meancur} is approximated by
 %
\begin{equation}
	U_n(z) = U_0 + \frac{\partial U}{\partial z} (z -  z_0),
\end{equation}
 %
with \mps{U_0=0.05}\ls{U_0}, \ps*{\partial U/\partial z=2.5}{-4}, and
\m{z_0=1900}\ls{z0}. Integrating $\Delta\theta_e(z)U_n(z)$ between
\m{1900} and \m{2300} across the plume width yields
\degCmmmps{Q_e\overline{\Delta\theta_e} \approx -230}\REM{efields -f
1900,5,2300 -r depth,spice '[$for]' '[2000*-0.019*0.25 *
exp(-(@depth@-2120)**2/(2*96**2))*(0.05+2.5e-4*(@depth@-1900))]' |
integrate -f spice depth \* \*}. The source temperature-anomaly flux
becomes \degCmmmps{Q_i\overline{\Delta\theta_i} \approx 550}\REM{factor
-2.42}, corresponding to a heat flux estimate of \GW{2.3}. (Using this
flux to force the bulk plume model of \rsecNP{HF-hor} results in a
predicted maximum rise height of \m{600}.) Considering the number of
assumptions and estimates the heat-flux value is based on it is
difficult to determine a confidence interval. In contrast to the
rise-height method, none of the parameters is raised to a power greater
than one, however, and those with the greatest uncertainties, namely
the width of the plume and the mean current speed, are most likely
accurate at least within a factor of two. Therefore, the lower and
upper bounds of the heat flux are estimated to be \GW{1} and \GW{5},
respectively.

