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%                    N E S P I / R E L A T I O N S . T E X 
%                    doc: Sun Aug 29 15:06:23 1999
%                    dlm: Wed Jun  7 23:29:54 2000
%                    uE-Info: 128 19 NIL 0 0 72 1 2 4 ofnI
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% SPICE ANOMALIES OF ISOPYCNAL INTRUSIONS

From a hydrographic perspective, hydrothermal equilibrium plumes are
neutrally buoyant intrusions with characteristic isopycnal temperature
and salinity anomalies $\Delta_\rho T$\ls{DrhoT} and $\Delta_\rho
S$\ls{DrhoS}. Neutral buoyancy requires the temperature anomalies to be
compensated by corresponding salinity anomalies, satisfying
 %
\begin{equation}
\alpha\Delta_\rho T = \beta\Delta_\rho S,
\label{eqn:compensation}
\end{equation}
 %
where $\alpha = \left. -\frac{1}{\rho} \frac{\partial\rho}{\partial T}
\right|_{S,p}$ and $\beta = \left. \frac{1}{\rho}
\frac{\partial\rho}{\partial S} \right|_{T,p}$. Therefore, the
hydrographic signatures of isopycnal intrusions can be characterized by
a single variable (the ``q'' of \citeN{misc/stommel62}) usually called
\emph{spice} or \emph{spiciness} \cite{inbook/munk81}. In the
following the isopycnal temperature anomaly of a water parcel will be
used to characterize its spiciness.

% T/RHO VS T/S

Calculating geothermal isopycnal anomalies as deviations from linear
background $T/\rho$ trends \cite{Nature/lupton+85} implies that the
deviations are caused solely by geothermal processes. To evaluate this
assumption it is useful to investigate the relationship between the
$T/\rho$ and the $T/S$ spaces. The typical rise height of hydrothermal
plumes (ignoring large event plumes) is a few hundred meters. The small
gradients of temperature and salinity characteristic of the water
column in the vicinity of deep hydrothermal vent fields indicate that
the hydrographic contrasts outside the buoyant plumes are small.
Therefore, the linear equation of state expanded about the reference
properties $S_0$, $T_0$, $p_0$ is a valid approximation, i.e.\
 %
\begin{equation}
\rho(S,T) = \rho_0 \left( 1 - \alpha_0 (T-T_0) + \beta_0 (S-S_0) \right),
\label{eqn:eqnstate}
\end{equation}
 %
with $\rho_0 = \left.\rho\right|_{S_0,T_0,p_0}$, $\alpha_0 =
\left.\alpha\right|_{S_0,p_0}$,\ls{alpha_0} and $\beta_0 =
\left.\beta\right|_{T_0,p_0}$.\ls{beta_0} The gradient of density with
respect to temperature is
 %
\begin{equation}
\frac{\partial\rho}{\partial T} = 
    \rho_0 
    \left( \beta_0
           \frac{\partial S}{\partial T} - \alpha_0
    \right),
\label{eqn:linear}
\end{equation}
 %
i.e.\ the relationship between the $T/S$ and the $T/\rho$ spaces is
linear, which implies the equivalence of $T/S$ linearity ($\partial
S/\partial T = \mathrm{const.}$) and $T/\rho$ linearity
($\partial\rho/\partial T = \mathrm{const.})$. 

% PROCESSES CHANGING LINEARITY

In addition to geothermal heating other processes, such as lateral
intrusions and differential diffusion of heat and salt, can change
linear $T/S$ relationships, resulting in non-geothermal isopycnal
hydrographic anomalies defined as deviations from linear background
$T/\rho$ relationships. (Lateral intrusions are common in the rift
valley of the \MAR, e.g.\ \rsecNP{F-waterorigin}.)

% EQUIVALENCE OF ISOPYCNAL AND ISOHALINE ANOMALIES

From expressions \reqn{compensation} and \reqn{eqnstate} it follows
that the constant factor relating the isopycnal to the corresponding
isohaline temperature anomalies ($\Delta_S T$\ls{DST}) observed in
regions of linear background $T/S$ relationships
\citeEG{JGR/thomson+95} can be calculated from the background
stratification of temperature and salinity. The constant of
proportionality is
 %
\begin{equation}
\frac{\Delta_S T}{\Delta_\rho T} =
    1 - \frac{\alpha_0}{\beta_0} \frac{\partial T}{\partial S} =
	1 - \Rr. 
\label{eqn:thomson+95}
\end{equation}
 %
Using the $T/S$ slopes given by \citeN{JGR/thomson+95} together with
the corresponding values for $\alpha_0$ and $\beta_0$ (evaluated at the
reference pressure of potential temperature and density) the constant
of proportionality becomes $0.52$ which compares well with the value of
$0.54\pm0.04$ calculated from their table~2, confirming that the linear
equation of state is a valid approximation in deep-sea hydrothermal
environments.

% RELATION BTW MCDOUGALL90/THOMSON+95

Combining \rexprs{mcd}{thomson+95} reveals that the isohaline
temperature anomalies associated with hydrothermal equilibrium plumes
rising from sources without salinity anomalies are equal to the source
temperature anomalies scaled by dilution. (The expressions can be
combined because they are both based on the same two assumptions,
namely neutral density and linear background $T/S$ relationships;
\De\theta of \rexpr{mcd} corresponds to $\Delta_\rho T$ of
\rexpr{thomson+95}.) Using spatially integrated isohaline temperature
anomalies to estimate the geothermal heat content of the water column
\citeEG{JGR/thomson+95}\REM{They translate the depth integral into the
salinity space by assuming a constant S_z slope but this does not
change the fact that they are using isohaline anomalies.} is therefore
equivalent to using isopycnal anomalies corrected with the expression
of \citeN{EPSL/mcdougall90} and assuming zero source salinity
anomalies.

% COLD/FRESH W/O ENTRAINMENT

\rExpr{thomson+95} implies that entrainment into buoyant plumes is not
required for the cold/fresh isopycnal anomalies observed in regions of
unstable background salinity stratification. If the bottom water is
conductively heated (and assuming that the heating is gradual enough so
that no convective instabilities develop) its density is reduced
without an associated change in salinity. Compared to background water
of the same density the geothermally modified water is fresh and
\rexpr{compensation} implies that it is cold. Entrainment and vertical
transport \emph{are} required for isobaric cold/fresh anomalies,
however.

