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%                    F L U X E S / H E I G H T - O F - R I S E . T E X 
%                    doc: Mon Feb  7 17:09:59 2000
%                    dlm: Wed Jun  7 23:56:41 2000
%                    (c) 2000 A.M. Thurnherr
%                    uE-Info: 110 32 NIL 0 0 72 3 2 4 ofnI
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A simple method for estimating heat fluxes from hydrothermal plumes is
based on source buoyancy flux estimates derived from height-of-rise
observations and more or less elaborate models of the buoyant plumes.
Even the most advanced of current numerical models cannot resolve
plumes from individual sources within a vent field and must rely on the
assumption of a single point (or line) source (i.e.\ the length scale
associated with the source area is assumed to be much smaller than the
plume rise height) \cite{JGR/lavelle97}. If this assumption is violated
the buoyancy flux estimated from the rise height of the model plume can
significantly underestimate the true buoyancy flux because the effects
of plume interaction at some height above the sources are ignored. The
individual vent chimneys of the \R hydrothermal field (located near the
\m{2300} isobath) are distributed over an area of
\m{100}$\times$\m{250} \cite{misc/fouquet+98}, indicating that the
extent of the integrated plume source is of the same order as the mean
rise height of \m{200} observed during the 1997 survey \rsec{P-1997}.

To estimate the buoyancy flux required to propel a plume to the
observed height-of-rise the buoyant time-averaged plume is modeled by
solving conservation equations for mass, heat, salt and momentum, i.e.\
 %
\begin{eqnarray}
\frac\partial{\partial z}(XW) 			& = & \Lambda X^{1/2}W \labelEqn{S+R1} \\ 
\frac\partial{\partial z}(TXW)			& = & \Lambda T_bX^{1/2}W \\
\frac\partial{\partial z}(SXW)			& = & \Lambda S_bX^{1/2}W \\
\rho_0\frac\partial{\partial z}(XW^2)	& = & g(\rho-\rho_b)X \labelEqn{S+R4}
\end{eqnarray}
 %
\citeEG{JGR/speer+rona89}, where $X$\ls{X} and $W$ are the
cross-sectional area and the vertical velocity of the plume, and
$T_b$\ls{T_b}, $S_b$\ls{S_b} and $\rho_b$\ls{rho_b} denote the
hydrographic properties of the background water column. The effects of
plume-generated turbulence are represented by an \emph{entrainment
assumption} which states that the amount of background fluid mixed into
the plume is proportional to the vertical plume velocity
\fullcite{RoySoc/morton+56}. The entrainment coefficient
$\Lambda=0.415$\ls{Lambda} is taken from \citeN{review/turner86}. A
linear equation of state (with \pdegC*{\alpha=1.47}{-4} and
\ppsu*{\beta=7.53}{-4} at \degC{4}, \dbar{2000} and \psu{35}, taken
from \citeN{books/gill82}) is used \cite{EPSL/turner+campbell87}. The
initial conditions are taken from the observed effluent properties of
the \R vent field \cite{misc/fouquet+98} and by choosing source
buoyancy fluxes --- in effect combinations of representative source
velocities (e.g.\ \mps{0.4}; \citeNP{JGR/speer+rona89}\REM{1m/s used
for all calculations as reasonable ord-of-mag estimate}) and areas ---
so that the maximum height reached by the model plume (where $W$
becomes negative) matches the observed height-of-rise. The approach
followed here is similar to that of \citeN{misc/rudnicki+elderfield92}
but individual nephelometry peaks are not assumed to be superimposed
plumes from separate sources \rchap{plume}.

Numerically solving equations \reqn{S+R1}--\reqn{S+R4}, using the mean
1997 SW basin hydrography for $T_b$, $S_b$ and $\rho_b$, results in a
source buoyancy flux estimate of \mmmmpsss{0.1} corresponding to the
observed maximum rise height of \m{400} \rsec{P-1997}. This yields a
heat-flux estimate of \GW{0.3} (using the low-temperature specific heat
\JpkgpK*{c_p=4.2}{3}; \citeNP{EPSL/turner+campbell87}). The total rise
time of the model plume is of order \h{1} and the temperature anomaly
at the level of neutral buoyancy is \degC{-0.05}, an order of magnitude
larger than the peak temperature anomalies associated with the Gaussian
near-field plume \rsec{HF-advective}. (Based on unpublished data from a
dye-release experiment the rise time of individual parcels of
hydrothermal plume fluid can be much shorter (of order minutes) than
the mean rise time predicted by bulk plume models; \perscommNP{A.\
Schultz}\emph{)}

Applying the height-of-rise method to the particle plume observations
of the 1998 survey (maximum rise height \m{500}; \rsecNP{P-1998}) and
using the corresponding hydrographic profiles results in a heat-flux
estimate of \GW{0.6}. The total rise time of the model plume is again
of order \h{1} and the temperature anomaly at the level of neutral
buoyancy is \degC{-0.04}. The apparent increase in heat flux, compared
to 1997, is not considered significant, primarily because the mean 1998
hydrography of the SW basin is derived from only 4 profiles and because
of the strongly sheared flow on the western slope of \RR bisecting the
plume \rsec{P-1998}. Given this strong shear it is not \emph{a-priori}
clear if the model of \fullciteN{RoySoc/morton+56} is applicable or if
background turbulence affects the plume rise as is often the case in
the atmospheric environment \citeEG{RoySoc/priestley56}.

The height-of-rise method as used here is closely related to estimating
the source buoyancy flux $B_i$\ls{B_i} from the maximum height
$z^*$\ls{zstar} attained by a plume rising from a point source into a
linearly stratified stagnant background, using
 %
\begin{equation}
 %
	z^* = 3.8 B_i^{1/4} N^{-3/4}						\labelEqn{turner}
 %
\end{equation}
 %
\citeEG{review/turner86}. The source buoyancy flux therefore depends on
the fourth power of the rise height and on the third power of the
buoyancy frequency, indicating that even in the absence of the problems
mentioned above, heat fluxes derived from buoyancy flux estimates based
on height-of-rise modeling are associated with large uncertainties
\citeEG{review/speer+helfrich95}.

% why Coriolis can be ignored safely and why cross flow is ignored

The effects of the Earth's rotation have been ignored in above
calculations. Direct effects on rising plume fluid can be disregarded
because of the short rise times which are small compared to the
inertial period of \h{20}. (Laboratory experiments of thermals rising
into stratified stagnant backgrounds indicate that rotational effects
can be ignored as long as $N/f<0.6$ \cite{JFM/helfrich94}; in the \A
segments \ps{N\approx10^{-3}} \rsec{sigma2}, implying that
$N/f\approx0.1$.) The numerical experiments of
\citeN{JMR/speer+marshall95} show that the rise height of starting
plumes can be influenced indirectly by rotational effects because of
the cyclonic flow set up around the buoyant plumes if the buoyancy
sources persist longer than an inertial period. Direct velocity
observations in the vicinity of the \R vent field (\rsecNP{LADCP} and
\rchapNP{CM}) indicate that the dynamics are dominated by the
along-valley flow without any indications for a plume-driven cyclonic
circulation. Therefore, it appears reasonable to assume that the
hydrothermal effluents rise into ``pristine'' background water carried
to the vent field by the along-valley flow, implying that rotational
effects on plume rise can be ignored.

\REM{The effects of the along-valley flow on plume rise have been
ignored as well because simple extension to the plume model of
\fullciteN{RoySoc/morton+56} which take (uniform) background flow into
account \citeEG{misc/middleton+thomson86} are based on the assumption
that the cross flow is strong enough to bend over the plumes to nearly
horizontal trajectories close to the buoyancy sources, something which
is not observed in video images of the \R vent field.}

