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%                    B O F / M O D E L . T E X 
%                    doc: Fri Mar 17 20:04:30 2000
%                    dlm: Thu Jun  8 00:20:32 2000
%                    (c) 2000 A.M. Thurnherr
%                    uE-Info: 89 0 NIL 0 0 72 3 2 4 ofnI
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\begin{figure}[t]

\placeFig{BOF/model}

\caption[Basin/sill model sketch] {Sketch of the \OneAndAHalfLayer
model consisting of a square basin of length \Bl and depth \Bd; the
model is forced by inflow \BQi into the lower layer; fluid can leave
the basin across an outflow sill (\BQo) and across the dashed
horizontal interface (\BQu); the interface height above the outflow
sill is denoted by \Bh.}

\labelFig{BOF-model}

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

\rFigNP{BOF-model} illustrates the simple 2-dimensional (along-valley
and vertical) \OneAndAHalfLayer (reduced-gravity) model intended to 
investigate the low-frequency along-valley flow within the \A segments.
The rift valley is represented as a flat-bottom basin of length
\lsym{Bl} and depth \lsym{Bd}. The model is forced by prescribing a
volume flux \lsym{BQi} into the lower layer. Fluid can leave the model
basin across an outflow sill (\lsym{BQo}) and across the horizontal
interface (\lsym{BQu}). The interface height above the outflow sill is
denoted by $\lsym{Bh}\geq0$. Ignoring thermal expansion, conservation of
volume becomes
 %
\begin{equation}
 %
\Bl \total{}{t}\Bh = \BQi - \BQu - \BQo.
												\labelEqn{BOF-volcons}
 %
\end{equation}
 %
The dynamics of the flow within the basin are not resolved, implying
that the model is only valid for time scales much longer than it takes
transients to propagate along the entire length of the basin. Using the
length (\km{\Bl=100}) and the mean depth below \m{2000} (\m{d=500}) of
the \sA segment \rfig{AMAR-bridge} as well as reduced-gravity estimates
from the hydraulic calculations at the inflow and at \RS
(\mpss*{\gp=0.5$--$1.8}{-4}; \rsecsNP{hydraulics}{F2-inflow}) yields a
time scale of \d{4$--$7} for long gravity waves to propagate between
the two sills. On longer time scales the interface adjusts
instantaneously to changes at the inflow. (The model does not support
slower waves such as higher-mode baroclinic gravity waves and
topographic Rossby waves (c.f.\ \rsecNP{CM-energy}), i.e.\ it is
assumed that the dynamically important transients in the rift valley
are Kelvin waves which propagate with the same speed as long
interfacial gravity waves \citeEG{books/gill82}.) 

The outflow \BQo is parameterized using the wide-sill hydraulic model of
\fullciteN{misc/whitehead+74} adapted to the 2-dimensional case, and
representing the effect of outflow across multiple sills by a scale
factor $\lsym{Bgamma}\geq1$, i.e.\
 %
\begin{equation}
	\BQo = \Bgamma\frac{\gp\Bh^2}{2 f\Bw},
													\labelEqn{BOF-w+74-2D}
\end{equation}
 %
where \lsym{Bw} is the width of the model basin. Substituting
\rexpr{BOF-w+74-2D} into \reqn{BOF-volcons} yields
 %
\begin{equation}
 %
\Bl\total{\Bh}{t} = \BQi - \BQu - \Bgamma\frac{\gp\Bh^2}{2\Bw f}.
													\labelEqn{BOF-model}
 %
\end{equation}
 %
In this framework the apparent volume-flux deficits of the \A segments
\rchap{mixing} can be represented as follows:\begin{description}

\item[volume flux is conserved] (i.e.\ the apparent deficits are caused
by uncertainties in the volume-flux budgets) $\Rightarrow$ $\BQu=0$,
$\Bgamma=1$;

\item[outflow across multiple sills] $\Rightarrow$ $\BQu=0$,
$\Bgamma=\Qi/\Qo$, where \lsym{Qi} and \lsym{Qo} are the observed
volume fluxes taken from \rtabNP{mix-budgets}, i.e.\ $\Bgamma=1.5$
\sAp{(}{)} and $\Bgamma=1.3$ \Ap{(}{);}

\item[``entrainment'' into the upper layer] $\Rightarrow$
$\BQu=(\Qi-\Qo)/\Bw$, $\Bgamma=1$, where \km{w=20} is the mean
rift-valley width at \m{2000} \rfig{AMAR-bridge}.

\end{description} \rExpr{BOF-model} implies that the effect of loss of
lower-layer fluid across the interface (i.e.\ $\BQi=\Qi/\Bw$,
$\BQu=(\Qi-\Qo)/\Bw$) is equivalent to the case where the model is
forced with a reduced inflow (i.e.\ $\BQi=\Qo/\Bw$, $\BQu=0$).
