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%                    N E S P I / M C D O U G A L L 9 0 . T E X 
%                    doc: Tue Aug  4 10:12:29 1998
%                    dlm: Thu Jun  8 00:39:11 2000
%                    (c) 1998 A.M. Thurnherr
%                    uE-Info: 108 61 NIL 0 0 72 3 2 4 ofnI
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Early hydrothermal plume studies were made in the Pacific where the
\emph{equilibrium plumes} (neutrally buoyant without vertical momentum)
are characterized by positive isopycnal and isobaric temperature
anomalies \citeEG{Nature/weiss+77}. Based on numerical experiments
\citeEG{EOS/mcduff88,JGR/baker+89} it was soon realized that the
relationship between the heat fluxes at the equilibrium level and at
the source is non-trivial because it depends on the background
hydrography as well as on the hydrothermal effluent properties.
Observation of negative temperature anomalies in the Atlantic
\cite{JGR/rona+speer89} prompted \citeN{JGR/speer+rona89} to
investigate this relationship. They showed that because of the unstable
salinity gradient in the deep Atlantic (e.g.\ \rfigNP{F-tscompare})
hydrothermal plumes rising from sources without salinity anomalies are
characterized by cold/fresh hydrographic anomalies.
\citeN{EPSL/mcdougall90} extended their work by taking variable source
salinities into account and derived an analytical expression for the
resulting isopycnal hydrographic anomalies. (In an earlier study,
\citeN{EPSL/turner+campbell87} had investigated the condition for
buoyancy reversal in rising plumes caused by the non-linearity of the
equation of state at high temperatures
\cite{misc/bischoff+rosenbauer85} to occur, finding that large positive
source salinity anomalies are required.) If the $T/S$ relationship of
the fluid entrained into the buoyant plume is linear, the
equilibrium-plume temperature anomaly flux
$Q_e\De{\theta}$\ls{Q_e}\ls{Debar} is related to the corresponding
source temperature anomaly flux $Q_i\Di{\theta}$\ls{Q_i}\ls{Dibar} by
 %
\begin{equation} 
	%
	\frac{Q_i\Di{\theta}}{Q_e\De{\theta}} =
		-\frac{\Rr-1}{1-\Rr/\Ri}
	\labelEqn{mcd}
	%
\end{equation}
 %
(from expression (22) of \citeNP{EPSL/mcdougall90}). $Q_i$ and $Q_e$
denote the volume fluxes at the source and in the equilibrium plume,
respectively; \Di\theta and \De\theta are the corresponding mean
temperature anomalies; $\Rr=\alpha\theta_z/(\beta S_z)$\ls{Rr} is the
stability ratio of the water column (with $\alpha$ and $\beta$ denoting
the linear thermal expansion and haline contraction coefficients
evaluated at the reference pressure of potential temperature,
respectively); $\Ri=\alpha\Di{\theta}/(\beta\Di{S})$\ls{Ri} is the
density-anomaly ratio of the hydrothermal source fluid (with
$\Di{S}$\ls{Sibar} denoting the mean effluent salinity anomaly). The
stable (unstable) salinity stratification in the deep Pacific
(Atlantic) implies $\Rr<0$ ($\Rr>0$). Even though
\citeN{EPSL/mcdougall90} assumes the vertical gradients of background
potential temperature and salinity to be constant, this is not used in
his derivation of \rexpr{mcd} --- it is sufficient to assume that \Rr
is constant, i.e.\ that the background $T/S$ relationship is linear.
(\citeN{DSR/lavelle+98} arrive at the same conclusion using a different
derivation.) Typical values for the stability ratio are $\Rr=-2$ in the
Pacific and $\Rr=2$ in the Atlantic (using the background
stratifications of \citeN{JGR/speer+rona89}, as well as
\pdegC*{\alpha=1.47}{-4} and \ppsu*{\beta=7.53}{-4} at \degC{4},
\dbar{2000} and \psu{35}, taken from \citeN{books/gill82}).

%--------------------
\begin{figure}[t]

\placeFig[1]{NESPI/bulkdiag}

\caption[Hydrothermal hydrographic-anomaly regimes]{Isopycnal
hydrographic-anomaly regimes of hydrothermal equilibrium plumes;
hydrographic background properties are characterized by the stability
ratio \Rr; effluent property anomalies are characterized by the
density-anomaly ratio \Ri; the parameters are not limited to the ranges
shown.}

\labelFig{bulkdiag}

\end{figure}
%--------------------

\rExpr{mcd} indicates that hydrothermal equilibrium plumes may be
cold/fresh or warm/salty independent of the sign of \Rr.
\rFigNP{bulkdiag} shows the possible hydrographic-anomaly regimes in
\Rr/\Ri space. In the hydrothermal context the source temperature
anomalies are positive by definition, implying that the sign of \Ri is
determined by the source salinity anomalies (salinity excesses implying
$\Ri>0$, with small values corresponding to large anomalies); the
plumes rising from sources with zero salinity anomalies investigated by
\citeN{JGR/speer+rona89} are represented by
$\Ri\rightarrow\!\pm\infty$.

The regime labeled ``Negative Source Salinity'' covers the unphysical
parameter range where the salinity deficit of the source fluid is
greater than the background salinity $S_b$, i.e.\ 
$-\alpha\Di{\theta}/(\beta S_b) < \Ri \leq 0$. The regime labeled
``Buoyancy Reversal'' ($0\leq\Ri\leq1$) is not covered by the plume
model because in this parameter range the linear equation of state
implies negative (or neutral) buoyancy at the source. The thermal
expansion coefficient $\alpha$ of the real equation of state is
significantly increased at the high temperatures typically observed in
hydrothermal sources \cite{misc/bischoff+rosenbauer85} so that the
effluents can initially be buoyant and reverse their buoyancy on mixing
with background sea water \cite{EPSL/turner+campbell87}.

The areas labeled ``cold/fresh'' and ``warm/salty'' cover the
hydrographic-anomaly regimes of hydrothermal equilibrium plumes. The
curves dividing the positive and negative \Rr half spaces into
cold/fresh and warm/salty regimes indicate $\Rr=\Ri$ where the
hydrographic anomalies vanish. (The sign of the equilibrium-plume
hydrographic anomalies is determined by the location of the
source-fluid density-anomaly ratio \Ri in relation to the linear
bisection of the $T/S$ space defined by the stability ratio of the
background water column (\Rr), because the $T/S$ properties of the
buoyant plume cannot cross the background properties as long as the
linear equation of state applies; see \citeN{EPSL/mcdougall90} for
details.) In addition to the ``classical'' Atlantic cold/fresh and the
Pacific warm/salty cases there are also Atlantic warm/salty
$(\Rr>\Ri>1)$ and Pacific cold/fresh
$(\Rr<\Ri\leq-\alpha\Di{\theta}/(\beta S_b)$ regimes. While these have
not been observed in the context of hydrothermal plumes the effluent
properties listed by \citeN{inbook/vondamm95} suggest that they may be
possible. (Cold/fresh anomalies have long been known to result from
``fresh'' sewage input into the ocean, independent of the sign of \Rr;
e.g.\ \citeN{misc/fischer71}.)

