%======================================================================
%                    M I X I N G / V A L L E Y . T E X 
%                    doc: Tue Apr  4 14:51:09 2000
%                    dlm: Thu Jun  8 00:12:39 2000
%                    (c) 2000 A.M. Thurnherr
%                    uE-Info: 180 39 NIL 0 0 72 3 2 4 ofnI
%======================================================================

To calculate heat budgets of the rift valley below \m{2000} the
corresponding mass (volume) budgets are required. Based on the
hydraulic volume flux estimates across sills ``I'' \rsec{F2-inflow} and
``R'' \rsec{hydraulics} the volume budget of \sA below \m{2000} is
characterized by inflow from the eastern ridge flank of \mmmps*{95}{3}
and outflow across \RS of \mmmps*{65}{3}, leaving a deficit of
\mmmps*{30}{3}. While outflow across sill ``O'' can balance this
budget, the hydrographic sections of \citeN{JGR/wilson+95} appear more
consistent with inflow. Furthermore, there are other potential sinks
for the rift-valley water entering \sA (see below).

Deriving a similar budget for \A is associated with even greater
uncertainties. The deepest sill near the northern end of the segment
connects the rift valley to the \FAM segment at \m{2200} (sill ``F'').
The  available bathymetry does not rule out an additional sill (sill
``W'') connecting \A to the western ridge flank at some depth above
\m{2100}. The northernmost rift-valley profiles of both hydrographic
surveys indicate higher densities at \m{2100} than in a western
ridge-flank background profile, with differences \kgpmmm{<\!10^{-2}}.
Applying the hydraulic model of \fullciteN{misc/whitehead+74} (c.f.\
\rsecNP{hydraulics}, with assumed sill and interface depths of \m{2100}
and \m{2000}, respectively) yields volume flux estimates
\mmmps*{<6}{3}, i.e.\ less than 10\% of the water entering \A across
\RS below \m{2000}.

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

\placeFig{mixing/outflow}

\caption[\A/\FAM sill observations]{Hydrographic and light-scattering
observations from sill ``F'' connecting the \A and \FAM segments; in
the main panel the potential density and nephelometry profiles are
plotted; the inset shows the corresponding \ThetaS properties.}

\labelFig{mix-outflow}

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

The most likely pathway out of the segment for the bulk of the water
flowing across \RS into \A is therefore across sill ``F''. (The
hydrographic sections of \citeN{JGR/wilson+95} appear consistent with
this view.) \rFigNP{mix-outflow} shows data from a station occupied
directly on that sill. Between \m{2020} and \m{2040} the density
profile is characterized by a steep gradient. The corresponding \ThetaS
properties rule out instrument problems and suggest that the water
above and below the density step is most likely of the same origin,
i.e.\ eastern North Atlantic water. (The slight change in the \ThetaS
slope near \degC{4.2} is consistent with other rift-valley profiles of
the 1998 survey, e.g.\ \rfigNP{tscompare}.) The available bathymetric
data sets indicate that the \FAM segments are closed below \m{2000}
with the exception of sill ``F'' and an additional sill on the eastern
rift-valley wall near \latN{36}{25}/\lonW{32}{55} with a minimum depth
between \m{2000} and \m{2100}. Inflow of eastern ridge-flank water into
\FAM and overflow across sill ``F'' into \A is therefore as consistent
with the \ThetaS observations of \rfigNP{mix-outflow} as is flow from
\A into \FAM. The increased light-scattering signals observed below the
density step suggest two layers of water, possibly flowing in different
directions. While no light-scattering data is available from \FAM the
observed nephelometry signal is consistent with the rift-valley
properties of the \A segments where nearly all nephelometry profiles
contain weak maxima of \V{0.01$--$0.02} near \m{2100}.

Attempting to infer the flow direction below \m{2000} across sill ``F''
from \rfigNP{sigsec} does not lead to unambiguous conclusions either.
The density patterns near the northern and southern ends of the \A
segments are similar, i.e.\ consistent with inflow from \FAM into \A.
The current-meter measurements on the other hand indicate northward
mean flow into the region of the upsloping isopycnal surfaces (c.f.\
\rsecNP{CM-flowstats}). Isopycnal uplifting upstream of \RS was
observed in 1997 \rsec{hydraulics} and there are indications for the
same effect in \rfigNP{sigsec}. Our data do therefore not allow to
infer a mean flow direction across sill ``F'' with confidence. It is
nevertheless tentatively assumed that the flow is from \A into \FAM,
primarily because added inflow into \A would require a large outflow,
presumably across sill ``W'', which does not appear consistent with our
density observations.

The volume flux across sill ``F'' is estimated with the hydraulic model
of \fullciteN{misc/whitehead+74} (c.f.\ \rsecNP{hydraulics}). No
cross-sill density section is available, implying that the interface
height and density difference between the two layers must be estimated
with an alternative method. Using the density step observed on the sill
\rfig{mix-outflow} to derive a reduced gravity and interface height
yields \mpss*{\gp=3}{-4} and \m{\hu=170} (from the altimeter at
\m{2030}, the center of the density step). The Rossby radius of
deformation corresponding to these values is \km{2.6}, less than the
sill width of \km{5} at \m{2000}, indicating that the hydraulic model
can be applied as before, yielding a volume flux estimate of
\mmmps*{50}{3}. The corresponding flow velocity estimated from the
interfacial wave speed is \mps{0.23}. The most likely volume budget for
the \A segment is therefore characterized by inflow across \RS of
\mmmps*{65}{3} and outflow into \FAM of \mmmps*{50}{3}, leaving a
deficit of \mmmps*{15}{3}.

It will be noted that the apparent volume flux deficits of the \A
segments are proportional to the lengths (or areas) of the segments.
While additional outflows across sills ``O'' and ``W'' can balance the
budgets it appears also possible that some of the rift-valley water is
lost to the overlying water column (see below). Using the horizontal
area of the \A segments at \m{2000} (calculated from the mean
rift-valley width of \km{20} at \m{2000} and the length of the
segments, i.e.\ \km{100} for \sA and \km{50} for \Ap{}{),} the apparent
volume-flux deficits imply a vertical velocity of \mps*{1.5}{-5}. (For
comparison, \citeN{JGR/read+pollard99} estimated a value of
\mps{10^{-5}} for the entire Mozambique basin.)

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

\begin{center}\begin{tabular}{|l||rrr|rrr|}
\hline%------------------------------------------------------------
	& \multicolumn{3}{c|}{\sA} & \multicolumn{3}{c|}{\A} \\
	& \multicolumn{1}{c}{$Q$}
		& \multicolumn{1}{c}{$\overline{\theta_2}$}
			& \multicolumn{1}{c|}{$\rho_0 c_p Q\overline{\theta_2}$}
				& \multicolumn{1}{c}{$Q$}
					& \multicolumn{1}{c}{$\overline{\theta_2}$}
						& \multicolumn{1}{c|}{$\rho_0 c_p Q\overline{\theta_2}$} \\
	& \multicolumn{1}{c}{[\mmmps{}]}
		& \multicolumn{1}{c}{[\degC{}]}
			& \multicolumn{1}{c|}{[\W{}]}
				& \multicolumn{1}{c}{[\mmmps{}]}
					& \multicolumn{1}{c}{[\degC{}]}
			            & \multicolumn{1}{c|}{[\W{}]} \\
\hline\hline%======================================================
Inflow & $95\times10^3$ & 3.634 & $1.45\times10^{12}$ &
		 $65 \times10^3$ & 3.777 & $1.03\times10^{12}$ \\
Outflow& $-65 \times10^3$ & 3.777 & $-1.03\times10^{12}$ &
		 $-50 \times10^3$ & 3.904 & $-0.82\times10^{12}$ \\
Deficit& $-30 \times10^3$ & 3.710 & $-0.47\times10^{12}$ &
		 $-15 \times10^3$ & 3.841 & $-0.24\times10^{12}$ \\
\hline%------------------------------------------------------------
Balance & & \multicolumn{2}{r|}{$4.79\times10^{10}$} & &
		    \multicolumn{2}{r|}{$3.08\times10^{10}$} \\
\hline%------------------------------------------------------------
\end{tabular}\end{center}

\caption[\A volume and heat budgets]{Volume and heat budgets of the \A
segments; volume fluxes ($Q$) are taken from sills ``I'', ``R'' and
``F''; average temperatures below \m{2000} ($\overline{\theta_2}$) are
taken from CTD profiles; ``Balance'' denotes the amount of heat
required to close the budgets (LHS of \rexpr{mix-saunders}).}

\labelTab{mix-budgets}

\end{table}
%--------------------

Combining the volume flux budgets of the \A segments with hydrographic
data from the 1998 survey, heat budgets are calculated as shown in
\rtabNP{mix-budgets}. The average temperatures of the overflows below
\m{2000} are taken from the CTD tow-yo across sill ``I''
\rfig{F2-inflow}, from the 1998 CTD stations occupied in the vicinity
of \RR \rfig{F2-stations}, and from the profile shown in
\rfigNP{mix-outflow}, respectively. The temperatures associated with
the volume flux deficits are set to the average values of the inflows
and outflows. From these heat balances diapycnal-diffusivity estimates
are calculated using \rexpr{mix-saunders} with
\degCpm*{\theta_z=1.3}{-3}, estimated from \m{100}-thick layers
centered at \m{2000} (c.f.\ \rsecNP{mix-saunders}). The known
geothermal contributions (\BHc and \BHp) again have a minor impact on
the budgets. The estimates for the diapycnal diffusivities become
\mmps*{\Kv=4.1}{-3} \sAp{(}{)} and \mmps*{\Kv=5.6}{-3} \Ap{(}{).}
Taking a conservative approach by assuming that the apparent volume
flux deficits are caused by overestimates of the inflowing volume
fluxes (which is more conservative than assuming that the deficits are
caused by additional outflows across sills ``O'' and ``W'') yields
\mmps*{\Kv=3.3}{-3} \sAp{(}{)} and \mmps*{\Kv=4.8}{-3} \Ap{(}{).} If
the flow below \m{2000} across sill ``F'' is from \FAM into \A no heat
budgets and corresponding diffusivities can be estimated for the \A
segment.

