%======================================================================
%                    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: Sun Mar 26 19:07:28 2000
%                    (c) 1998 A.M. Thurnherr
%                    uE-Info: 86 65 NIL 0 0 72 3 2 4 ofnI
%======================================================================

Using the assumptions of the buoyant plume model of
\citeN{RoySoc/morton+56}, bulk temperature anomalies of neutrally
spreading plumes due to buoyant fluid rising into an environment with
linear potential temperature ($\theta$) and salinity ($S$)
stratifications, are

\begin{equation}
\De{\theta} = \left(\frac{Q^i}{Q^e}\Di{\theta}\right)
			  \times\left(\frac{1}{1-\Rr}\right)
			  \times\left(1-\frac{\Rr}{\Ri}\right).
\label{eqn:dilution}
\end{equation}

The notation used in \reqn{dilution} and below is as follows. The
superscripts denote properties of the background water column at source
depth ($^b$), of the hydrothermal source fluid ($^i$), of the
background water column at the equilibrium depth ($^e$, defined as the
depth where the neutrally buoyant plume spreads), at the depth where
the plume first becomes neutrally buoyant ($^n$), and at the maximum
rise height ($^*$). $\Delta^y X$ is defined as the excess of property
$X$ at level $y$ of the bulk plume fluid relative to the background
water column. $Q$ is a volume flux, $\Rr=(\alpha\theta_z)/(\beta S_z)$
the stability ratio of the water column (the $_z$ subscripts denote
vertical derivatives; $\alpha=1.47\times10^{-4\circ}\mathrm{K}^{-1}$
and $\beta=7.53\times10^{-4}\mathrm{psu}^{-1}$ are the linear thermal
expansion coefficient and the haline contraction coefficient at
\degC{4}, \dbar{2000}, and \psu{35} taken from
\citeNP{books/gill82}). Typical values for \Rr\ are -2.0 in the
Pacific and 2.0 in the Atlantic (using the background stratifications
of \citeNP{JGR/speer+rona89}). $\Ri =
(\alpha\Di{\theta})/(\beta\Di{S})$ is the density anomaly ratio of the
input fluid which varies according to the relative contributions of
temperature and salinity to the density deficit at the source. Both
$R_\rho$ and $R_\rho^i$ are strongly dependent on $\alpha/\beta$
indicating that care has to be taken to select values in the range of
the environmental conditions.

Note that $z^*>z^e>z^n$ due to the momentum of the plume plume at $z^n$
and the continuing entrainment --- the plume continues to rise to $z^*$
where it becomes negatively buoyant and drops back to the  equilibrium
depth $z^e$. (Plume trajectories in stratified crossflow in laboratory
experiments can be seen to oscillate in a damped wave-like motion
asymptotically approaching $z^e$, \citeEG{books/fischer+79}.)

The formal requirement for \reqn{dilution} to hold is that \Rr\ is
constant which is a weaker requirement than linear background
temperature and salinity stratifications. Such a uniform stability
ratio stratification is characterised by a linear \ThetaS\
relationship, a good approximation for the rift-valley stratification
at \R. \Ri\ is proportional to the slope of the \ThetaS\ line which
connects the hydrothermal source fluid properties with the background
properties at the source depth.

\citeN{EPSL/mcdougall90} re-writes \reqn{dilution} to yield

\begin{equation}
\De{\theta} = -\frac{\theta^e-\theta^b}{3.81\Rr}
			  \times\left(1-\frac{\Rr}{\Ri}\right)
			  \times\left(1-\frac{1}{\Ri}\right)^{-1}.
\label{eqn:mcdougall90}
\end{equation}

The constant $3.81$ depends on the value of the entrainment coefficient
of the \citeN{RoySoc/morton+56} model, the expression relating $z^e$ to
the \emph{asymptotic buoyancy flux} \cite{EPSL/turner+campbell87}, as
well as the volume flux at the equilibrium height $Q^e$.

\citeN{EPSL/mcdougall90} sets $Q^e=Q^*$, i.e.\ he assumes that the
entrainment assumption holds up to the maximum height of rise after
which entrainment stops. The plume cap observations of
\citeN{misc/abraham+eysink69} on the other hand suggest that $Q^e<Q^*$,
indicating that expression \reqn{mcdougall90} is a lower bound for the
bulk temperature anomalies. As the derivation of expression
\reqn{dilution} is based on the assumption of neutral buoyancy, an
analogue expression can be derived for $\Dn{\theta}$ using $Q^n$
instead of $Q^e$. Using the values of Table~1 in
\citeN{RoySoc/morton+56}, $Q^n/Q^e\approx0.7$, indicating that
$\Dn{\theta}/\De{\theta}\approx1.45$ (see \rtabNP{results} below). Due
to continued entrainment above $z^n$ this an upper bound for the bulk
temperature anomalies. Therefore, the real anomalies are expected to
lie somewhere between $\Dn{\theta}$ and $\De{\theta}$.

An alternative form of expression \reqn{mcdougall90} based on $z^*$
instead of $z^e$ can be derived as well. This is useful if the maximum
height-of-rise is known rather than the equilibrium height. The
constant for this case is $4.86$.

%----------------------------------------------------------------------

\Subsubsection{NESPI-ts}{\ThetaS\ Characteristics of the Rising Plume}

Derivation of expressions \reqn{dilution} and \reqn{mcdougall90} is
based on the conservation of $Qq$ in the rising plume, where 

\begin{equation}
q(\theta,S) = (\theta-\theta^b) - \frac{\theta_z}{S_z}(S-S^b)
			= (\theta-\theta^b) - \frac\beta\alpha\Rr(S-S^b).
\label{eqn:q}
\end{equation}

As long as \Rr\ is constant, $q$ is a linear combination of
conservative variables and, thus, conservative itself. On a \ThetaS\
diagram $q$ isolines lie parallel to the background \ThetaS\ where
$q=0$. The value of $q$ defines a ``distance'' between the plume and
the background properties. The sign of $q$ determines on which side of
the background \ThetaS\ a given point lies.

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

\placeFig{NESPI/plumets}

\caption[\ThetaS\ trajectory of a buoyant plume in the
Atlantic]{\ThetaS\ trajectory of a buoyant plume in the Atlantic; at a
given time the \ThetaS\ characteristics of the plume and background at
the same height are $P$ and $Q$ respectively, mixing between the two
determines the instantaneous trajectory of the plume \ThetaS\
($P\rightarrow Q$). At some later stage the respective points are $P'$
and $Q'$. At the equilibrium height the \ThetaS\ of the plume and
background are $E$ and $B$. The dashed line denotes the isopycnal of
the spreading plume. The anomaly associated with $E$ is cold/fresh
while $E'$ is an example of an isopycnal warm/salty anomaly.}

\labelFig{plumets}

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

\ThetaS\ trajectories of rising plumes have been studied by a number of
people
\cite{misc/middleton+thomson86,EPSL/turner+campbell87,EOS/mcduff88,JGR/speer+rona89,EPSL/mcdougall90}.
Except for initial non-linear mixing effects due to the high
temperatures of hydrothermal effluents \cite{EPSL/turner+campbell87}
the trajectories are straightforward. At any given time the plume fluid
is undergoing mixing with background water as shown in
\rfigNP{plumets}. The plume's \ThetaS, in the words of
\citeN{EPSL/mcdougall90}, ``is chasing a moving target''
(represented by $Q\rightarrow Q'\rightarrow B$). Once the plume has
reached its equilibrium height it spreads on an isopycnal (denoted by
the dashed line connecting $E$ and $B$). In the example shown, the
neutral plume is characterised by a cold/fresh anomaly. For a
warm/salty anomaly the point $E$ would have to lie on the other side of
the background \ThetaS\ (e.g.\ at $E'$). The sign of the anomaly is
determined by the sign of $q^e$. Given a straight background \ThetaS\
(constant \Rr) the plume's trajectory cannot cross the background
\ThetaS\ (this follows directly from the conservation of $Qq$).
Therefore, the sign of the hydrographic anomaly of the plume is
determined by the properties of the source fluid (i.e.\ the sign of
$q^i$). Using \reqn{q} and the definitions of \Rr\ and \Ri\ allows the
following expression to be derived:

\begin{equation}
q^i = \Di{\theta} - \frac\beta\alpha\Rr\Di{S}
	= \Di{\theta} \left( 1 - \frac{\Rr}{\Ri} \right) .
\label{eqn:qi}
\end{equation}

Note that $\Rr=\Ri$ indicates that $q^i=q=0$ (i.e.\ the plume \ThetaS\
lies on the straight line defined by the background \ThetaS) resulting
in zero hydrographic bulk anomalies of the spreading plumes (the
``phantom plumes'' of \citeNP{review/mcduff95}) \reqn{qi} also indicates
that, at least in theory, cold/fresh and warm/salty anomalies are
possible for any value of $\Rr\neq 0$, i.e.\ both in the deep Atlantic
($\Rr>0$) and Pacific ($\Rr<0$).

%----------------------------------------------------------------------

\subsubsection{The Bulk Anomaly Diagram}

The observations of the preceding section indicate that the established
view of cold/fresh hydrothermal plumes in the Atlantic and warm/salty
ones in the Pacific may not convey the full picture. To investigate the
different regimes in a systematic way expression \reqn{dilution} is
analysed.

For hydrothermal plumes the first bracketed term is always positive as
$\Di{\theta}>0$ by definition. Its value is the temperature anomaly
which would be the consequence of the same amount of dilution
($Q^e/Q^i$) of a plume rising into an unstratified environment. Typical
values for the dilution in neutrally spreading hydrothermal plumes are
of order $10^4$ \cite{Nature/lupton+85} indicating that the bulk
anomalies are of order \degC{10^{-2}} as the other terms of
\reqn{dilution} are of order unity.

The second bracketed term of \reqn{dilution} changes sign when $\Rr=1$.
As temperature is stably stratified in both the deep Pacific and
Atlantic oceans, $0<\Rr\leq1$ indicates static instability due to
unstable salinity stratification, something which is not observed in
the oceans. Therefore, the sign of this term is determined by the sign
of $\theta_z$, positive in the Pacific ($\Rr<0$) and negative in the
Atlantic ($\Rr>1$).

The sign of the third bracketed term of \reqn{dilution}  is determined
by the magnitude of $\Rr/\Ri$ --- this ratio represents the relation
between the hydrothermal source fluid and the background stratification
as outlined in \rsecNP{NESPI-ts}.

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

\placeFig{NESPI/bulkdiag}

\caption[Hydrographic bulk anomaly diagram]{Bulk anomaly diagram;
background stratification is represented by the stability ratio \Rr;
source fluid property anomalies are represented by the density anomaly
ratio \Ri. The hatched areas denote regions in the parameter space
where the model is not applicable, while the six unhatched regions
marked by roman numerals denote the six different possible regimes (see
text).}

\labelFig{bulkdiag}

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

\rFigNP{bulkdiag} shows the different regimes as unhatched regions in a
diagram of \Rr\ vs.\ \Ri. In the case of hydrothermal plumes the source
temperature anomalies are positive by definition, indicating that the
sign of \Ri\ is determined by the sign of the source salinity anomaly,
a salinity excess indicating a positive value. Note that the larger the
salinity anomaly, the smaller the value of \Ri; the zero salinity
anomaly used by \citeN{JGR/speer+rona89} is represented by
$\Ri\!\rightarrow\!\pm\infty$ which is the reason behind the statement
of \citeN{EPSL/mcdougall90} that the bracketed terms in
\reqn{mcdougall90} are often close to unity.

The crosshatched area of \rfigNP{bulkdiag} covers the impossible regime
where the salinity deficit of the source fluid is greater than the
background salinity, requiring a negative-salinity source fluid. The
parameter range for this physically impossible regime is
$-(\alpha\Di{\theta})/(\beta S^b) < \Ri\leq0$.

The diagonally hatched regime ($0\leq\Ri\leq1$) is physically possible
but not covered by the model. In this regime, the source fluid is
initially less dense than the background water (due to the temperature
excess) but reverses its buoyancy on mixing due to a large salinity
excess \cite{EPSL/turner+campbell87}. Note that the linear
equation of state for low temperatures describes the eventual buoyancy
reversal --- to determine if the fluid is initially buoyant a much
higher value for $\alpha$ valid near the high temperature measured at
the source must be used \cite{misc/bischoff+rosenbauer85}.

The vertically hatched region covers the unrealistic regime where the
unstable salinity stratification dominates the stable temperature
stratification rendering the water column statically unstable (see
above).

The line $\Rr=\Ri$ further subdivides the parameter space of a source
with a salinity excess in the Atlantic respectively a source with a
salinity deficit in the Pacific. It follows from the definitions of
\Rr\ and \Ri\ that this is the locus of source characteristics which
lie on the (linear) background \ThetaS\ and thus have no associated
neutral plume anomalies \rsec{NESPI-ts}.

Each of the six unhatched regions of \rfigNP{bulkdiag} represents a regime
where the sign of the anomalies does not change:

\renewcommand{\theenumi}{\Roman{enumi}}\begin{enumerate}

\item ($\Rr<0$, $\Ri>1$) This is the ``classic'' Pacific regime with a
source fluid salinity excess. The anomaly of the neutrally spreading
plume is warm/salty \cite{JGR/speer+rona89}. This is immediately
apparent from the fact that salinity decreases upwards in the deep
Pacific ocean --- at the neutral depth the plume is a mixture between
source fluid (salinity excess) and water from depths below the
spreading depth (salinity excess).

\item ($\Rr>1$, $\Ri>\Rr$) This is the ``classic'' Atlantic regime with
a source fluid salinity excess. The anomaly of the neutrally spreading
plume is cold/fresh \cite{JGR/speer+rona89}.

\item ($\Rr>1$, $1<\Ri<\Rr$) This is the ``new'' Atlantic regime
characterised by warm/salty plume anomalies. It is only possible for
large source salinities. The hydrothermal source fluid properties
listed by \citeN{inbook/vondamm95} suggest that this regime might be
possible.

\item ($\Rr>1$, $\Ri<-(\alpha\Di{\theta})/(\beta S^b)$) This regime is
the natural continuation of regime II for Atlantic sources with
salinity deficits. The anomalies are cold/fresh as the spreading plume
is a mixture of source and entrained background fluids, both of which
are fresh in comparison to the background fluid at equilibrium height. 

\item ($\Ri<\Rr<0$) This is the natural continuation of regime I for
Pacific sources with salinity deficits. The sign of the equilibrium
level bulk anomalies is warm/salty.

\item ($\Rr<\Ri<0$) This is the ``new'' Pacific regime with cold/fresh
plume anomalies. It is not a priori apparent if this regime is possible
at all (depending on the observed values of \Rr\ and \Ri\ in the
Pacific). For $0>\Rr>-(\alpha\Di{\theta})/(\beta S^b)$ this regime does
not exist as reducing the source salinity to zero is not enough to
cause cold/fresh anomalies. The hydrothermal source fluid properties
listed by \citeN{inbook/vondamm95} suggest that this regime might be
possible.

\end{enumerate}

%----------------------------------------------------------------------

\Subsubsection{NESPI-quant}{The Magnitude of the Hydrographic Anomalies}

\REM{%--------------------
\begin{figure}

\placeFig{contours}

\caption[Non-dimensional bulk temperature anomaly
contours]{Non-dimensional bulk temperature anomaly contours of
expression \protect\reqn{dilution}; shading denotes cold/fresh regime;
contour interval is 0.001.}

\labelFig{bulkdiag-contours}

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

To determine the relative magnitude of the expected bulk temperature
anomalies of Atlantic vs.\ Pacific plumes expression \reqn{dilution} is
analysed. ``Typical'' hydrothermal sources are characterised by small
source salinity anomalies, indicating that $\Ri\rightarrow\pm\infty$
\cite{EPSL/mcdougall90} so that the third term of \reqn{dilution} tends
to unity and can be dropped. While $Q^i$ and $\Di{\theta}$ can be taken
as equal for ``typical'' Atlantic and Pacific hydrothermal sources, any
difference of the bulk anomalies is due to the background
stratification which affects both $Q^e\propto N^{-5/4}$
\cite{EPSL/mcdougall90} and \Rr (slope of the background \ThetaS).
Using the linear equation of state,
$N^{-5/4}\propto(\alpha\theta_z-\beta S_z)^{-5/8}$ which, together with
the values (background stratification and source fluid properties) from
\citeN{JGR/speer+rona89} leads to $Q^e_\mathrm{Atlantic} /
Q^e_\mathrm{Pacific}\approx3.6$. \REM{3.599209} The respective ratio of
the second term of \reqn{dilution} is $-3$ indicating that
$\Delta^e\theta_\mathrm{Atlantic} / \Delta^e\theta_\mathrm{Pacific} 
\approx-0.85$. \REM{-0.8335162} For these parameter values the expected
bulk anomalies are slightly smaller (and of opposite sign) in the
Atlantic than in the Pacific.

\REM{\Fig{quant}{Relative expected temperature anomalies for variable
source conditions $R_\rho(SRP)=-2.0$, $R_\rho(SRA)=2.0$,
$Q^e(SRA)=3.6Q^e(SRP)$ where \emph{SRA} and \emph{SRP} denote typical
Atlantic and Pacific background conditions taken from
\protect\citeN{JGR/speer+rona89}.}}

The relative magnitudes can be arbitrarily large for certain values of
\Ri. For $\Ri=2$ the Atlantic plume has zero anomalies while the same
is true for the Pacific plume with $\Ri=-2$. No values have been
plotted for $0\leq\Ri\leq1$ (buoyancy reversal) and for
$0>\Ri\geq-1.672$ (crosshatched region in \rfigNP{bulkdiag} with
\psu{S^b=35}). For the \R site, for example, $\Ri\approx5.4$
\cite{misc/fouquet+98} indicating that the expected bulk temperature
anomaly is less than half the one for a similar source in the Pacific.

%----------------------------------------------------------------------

\subsubsection{Regime Modeling}

\begin{table}[ht]
\begin{center}\begin{tabular}{|l|rrr|}
\hline
Regime	&	$R_\rho$	&	$R_\rho^i$	&	$S^i$ [psu]	\\
\hline\hline
SRP (I\&V)	 &	-2.0	&	$\pm\infty$	&	34.608	\\
SRA (II\&IV) &	 2.0	&	$\pm\infty$	&	34.923	\\
III		&		 2.0	&	 1.85		&	2.5 \\
VI		&		-2.0	&	-1.85		&	67.0 \\
\hline
\end{tabular}\end{center}

\caption[Temperature anomaly model parameters]{Model parameters; linear
equation of state
$\rho=1036.903\times(1-1.469\times10^{-4}(\theta-4)+7.532\times10^{-4}(S-35)$
\protect\cite{books/gill82}; Atlantic/Pacific background profiles
$\theta_A(z)=2.35+4\times10^{-4}z$, $S_A(z)=34.923+4\times10^{-5}z$,
$\theta_P(z)=1.8+10^{-3}z$, $S_P(z)=34.608-10^{-4}z$, source
temperature $\theta^i=300^\circ\mathrm{C}$, entrainment coefficient
$E=0.332$,  source area and velocity $A^i=0.1\mathrm{m}^2$,
$w^i=0.4\mathrm{ms}^{-1}$ \protect\cite{JGR/speer+rona89}.}

\label{tab:params}
\end{table}

\rTabNP{params} lists the modelling parameters used. The \emph{SRA} and
\emph{SRP} regimes use the parameters (background hydrography and plume
source anomalies) from \citeN{JGR/speer+rona89} while the values of \Ri
(and, thus, $S^i$) for regimes III and VI have been selected by keeping
the same source temperature (\degC{\theta^i=300}) and selecting source
salinity anomalies ($\Di{S}=\pm32.4\mathrm{psu}$) to lead to effluent
salinities lying within the range (and near the extreme values) of
observed values given by \citeN{inbook/vondamm95}. Setting
$R_\rho=R_\rho^i$ allows the minimum salinity anomaly required for
regimes III and IV to be calculated. As
$\Di{S_{min}}=\pm30\mathrm{psu}$ the parameter range for those regimes
is rather narrow.

%----------------------------------------------------------------------

\begin{figure}[t]

\placeTwoFigs[0.6]{NESPI/aTS}{NESPI/pTS}

\caption[Temperature anomaly model \ThetaS evolution]{\ThetaS\
evolution for the modelling parameters of \protect\rtabNP{params}.}

\labelFig{regimes-model}

\end{figure}

\rFigNP{regimes-model} shows the \ThetaS\ evolution of the model runs
of \rtabNP{params}. The closeness of the initial slopes of the plume
\ThetaS\ for the ``new'' regimes to the background \ThetaS\ slopes
which is due to the high temperature of hydrothermal effluents
indicates that the associated hydrographic anomalies will be small.

%----------------------------------------------------------------------

\subsubsection{Modeling}

\begin{table}[ht]
\begin{center}\begin{tabular}{|l|rrr|rr|}
\hline
Regime
	& $z^n$ [m] & $\Dn{\theta}$ [\degC] & $\Dn{S}$ [psu]
	& $\theta^n-\theta^b$ [\degC] & $\De{\theta}$ [\degC] \\
\hline\hline
%Regime		 & z^n	& Dt_m	 & DS_m		& t^n-t^b	&	Dt_90
%----------------------------------------------------------------------
SRP (I\&V)   & 137	&  0.027 &  0.0053 	& 0.137 	&  0.018 \\
SRA (II\&IV) & 296	& -0.024 & -0.0046 	& 0.118 	& -0.016 \\
III			 & 242	&  0.003 &  0.0006 	& 0.097 	&  0.002 \\
VI			 & 154	& -0.002 & -0.0002 	& 0.154 	& -0.001 \\
\hline
\end{tabular}\end{center}

\caption[Comparison of theoretical and modelled temperature anomaly
values]{Comparison of model results and values calculated using
\protect\reqn{mcdougall90}.}

\label{tab:results}
\end{table}

\rTabNP{results} lists the equilibrium bulk temperature anomalies
$\De{\theta}$ calculated from expression \reqn{mcdougall90} and the
neutral level bulk temperature anomalies observed during the model runs
(see \rsecNP{expr} for the relation between \Dn{\theta} and
\De{\theta}). The anomalies of the ``new'' regimes are an order of
magnitude smaller than those of the ``classic'' regimes indicating that
they might be very hard to observe in the real ocean.

$\De{\theta}$ can be extracted from the model output in the following
way:

\begin{itemize}

\item Find the plume density at $z^*$ (\kgpmmm{1036.9} for SRP).
\item Find the level where the plume fluid of this density spreads
      (\m{z^e=143}).
\item The temperature anomaly is now the difference between the plume
      temperature at $z^*$ (\degC{1.9622}) and the background temperature
	  at $z^e$ (\degC{1.943}).

\end{itemize}

