%======================================================================
%                    C M / T R A N S P O R T . T E X 
%                    doc: Mon Mar  6 22:12:27 2000
%                    dlm: Wed Jun  7 23:06:16 2000
%                    (c) 2000 A.M. Thurnherr
%                    uE-Info: 206 0 NIL 0 0 72 3 2 4 ofnI
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\Subsection{CM-flowstats}{Flow Statistics}

\input{CM/tab_stats}

% main instruments to discuss

To assess the characteristics of the along-segment flow a number of
statistics were calculated as shown in \rtabNP{CM-flowstats} for
selected instruments. Attention is restricted to depths below \m{2000}
at the overflow (moorings B and D), on the western flank of \RR
(mooring G, \km{3} upstream of \RSp{}{),} and \km{30} downstream of \RS
(mooring F). In most records the mean flow direction was similar before
and after the warming with some backing observed during the event.
During its passage the mean flow across \RS appears to have shifted
upwards in the water column, i.e.\ it increased at \m{2100} and
decreased at \m{2300}. The data from mooring F are consistent with
bottom-intensified continuation of the unidirectional flow along the \A
segments (c.f.\ \rchapNP{mixing}). The smaller velocities measured
there most likely indicate that the flow is of greater lateral extent.
The along-mean (\lH) and cross-mean (\cH) standard deviations do not
show any striking patterns. \REM{Mooring F shows the lowest
along-stream standard deviation while mooring G shows the highest one.
The cross-stream variabilities are more uniform. During passage of the
warming event the along-mean variability is increased everywhere except
at mooring G where both along- and cross-mean values are decreased. At
the remaining moorings the cross-mean variabilities are decreased at
\m{2300} and increased at \m{2100}. The post-warming variability is
higher than those of both previous regimes at most instruments except
at mooring G where it is very similar to the pre-warming situation.}

The reliability of the mean velocity estimates of \rtabNP{CM-flowstats}
depend on the spectral characteristics of the flow. If there are
energetic low-frequency oscillations which are not adequately resolved
the calculated mean values may not be representative of the long-term
averages. To estimate the energetic time scales of the flow the
integral time scales $\tau_l$\ls{taul} are calculated by integrating
the lagged autocorrelations of the low-pass filtered (to remove tidal
fluctuations) along-mean velocity components to the first zero
crossings \citeEG{JPO/saunders87}. \citeN{JMR/flierl+mcwilliams77} show
that for red spectra the expected error variance of the mean is
(conservatively) approximated by
 %
\begin{equation}
 %
\epsilon\ls{epsilon} = \frac{2\tau_l}{T_r\ls{Tr}}\sigma_l^2.
\label{eqn:meanerr}
 %
\end{equation}

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

\begin{figure}[t]

\placeFig{CM/B23autocorr}

\caption[\RS along-mean-flow autocorrelations]{Lagged autocorrelations
of the pre-warming along-mean velocity component recorded by instrument
B23; correlations for lags greater than half the record length are not
shown; the ``unfiltered'' autocorrelation is derived from all spectral
components with frequencies greater than \cpd{1} while the frequencies
in the IF band \protect\rsec{CM-energy} were removed from that range to
derive the ``filtered'' autocorrelation.}

\labelFig{CM-autocorr}

\end{figure}

%--------------------
\input{CM/tab_overrs}
%--------------------

\rFigNP{CM-autocorr} shows the autocorrelation of the pre-warming B23
along-mean flow (labeled ``unfiltered''). The deterministic oscillation
with a frequency of \cpd{0.23} indicates that the spectrum is peaked in
violation of the assumptions underlying \rexpr{meanerr}. Because
sub-diurnal oscillations are present in many of the current meter
records \rsec{CM-spectra} these were removed by filtering the IF band
\rtab{CM-fbands} from the ``contaminated'' records\REM{pre: BD23,
event: BD23, post: D23}. The resulting ``filtered'' autocorrelation is
shown in \rfigNP{CM-autocorr} as well. Integrating the (filtered where
necessary) autocorrelations yields the integral time scales listed in
\rtabNP{CM-ovvel} together with the mean flow velocities and the
associated error estimates derived from \rexpr{meanerr}. The integral
time scales are relatively uniform at \m{2300} throughout the sampling
period while there are large variations at \m{2100}. (For comparison,
\citeN{JPO/saunders87} reports values around \d{3} derived from flow
measurements characterized by similar statistics in Discovery Gap.)

%======================================================================

\Subsection{CM-volflux}{Volume Flux Across \RS}

\begin{figure}[t]

\placeFig{CM/fluxcalc1}

\caption[\RS transport calculation (plan view)]{Plan view illustrating
the \RS volume-flux calculations; white arrows show arbitrarily scaled
pre-warming current meter measurements at \m{2300}; black arrows show
the corresponding translated and projected\REM{48 degrees from N}
velocities; the heavy straight line indicates the along-sill section
(c.f.\ \protect\rfigNP{CM-fluxcalc2}); bathymetric shading is the same
as in \protect\rfigNP{AMAR-bridge}; contour interval is \m{100}; the
heavy contour marks the \m{2300} isobath.}

\labelFig{CM-fluxcalc1}

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

\placeFig{CM/fluxcalc2}

\caption[\RS transport calculation (cross-valley section)]{\RS
cross-valley section (heavy line in \protect\rfigNP{CM-fluxcalc1});
distance is measured from \RR; in addition to the instrument labels
printed at the locations of the current meters the cross-sectional
surface areas of the four boxes (in units of \mm{10^3}) are given in
parenthesis.}

\labelFig{CM-fluxcalc2}

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

\begin{center}\begin{tabular}{|l||r|r|r|}
\hline%----------------------------------------------------------------------
		& \multicolumn{1}{c|}{pre-warming}
			& \multicolumn{1}{c|}{warming}
				& \multicolumn{1}{c|}{post-warming} \\ 
\hline\hline%======================================================================
best  &	\mmmps*{73.2(\pm7.7)}{3} & \mmmps*{65.6(\pm14.4)}{3} & \mmmps*{71.2(\pm18.4)}{3} \\
upper &	\mmmps*{92.1(\pm9.7)}{3} & \mmmps*{73.1(\pm16.2)}{3} & \mmmps*{84.3(\pm21.2)}{3} \\
lower &	\mmmps*{50.4(\pm4.6)}{3} & \mmmps*{43.9\ (\pm9.1)\ }{3}  & \mmmps*{48.7(\pm11.7)}{3} \\
\hline%----------------------------------------------------------------------
\end{tabular}\end{center}

\caption[\RS transport]{\RS volume-flux estimates.}

\labelTab{CM-volflux}

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

The cross-sill velocities of \rtabNP{CM-ovvel} are now used to
calculate volume-flux estimates for the \RS overflow below \m{2000}. To
this purpose a section along the sill is defined between the northern
tip of the \m{2000} bathymetric contour of \RR and the western
rift-valley wall, mooring B is translated along isobaths onto that
section, and the velocities are projected on the cross-sill direction
(\deg{48}). These steps are illustrated with the pre-warming velocities
at \m{2300} in \rfigNP{CM-fluxcalc1}. The cross-valley section below
\m{2000} is divided into 4 boxes as illustrated in
\rfigNP{CM-fluxcalc2}. There is one velocity estimate in each box which
is taken to represent its mean velocity. The resulting transport
estimates across \RS are listed in the first row of
\rtabNP{CM-volflux}, labeled ``best''.

In addition to the error bars of the transport estimates derived from
the errors associated with the mean velocities \rtab{CM-ovvel}, there
are additional uncertainties because of the small spatial sampling
resolution and the details of the flux calculation. It can be argued
that the flow recorded on mooring B should be anticlockwise rotated
before projection because of possible flow curvature (following the
isobaths) between the true and the translated mooring positions
\rfig{CM-fluxcalc1}. The appropriate amount of rotation is difficult to
estimate but an upper bound for the transport (assuming that there are
no other effects low-biasing the estimates) can be derived by repeating
above calculations using the along-mean velocities of mooring B without
projection. The resulting volume-flux estimates are listed in the
second row of \rtabNP{CM-volflux}, labeled ``upper''.

The assumption that each velocity estimate is representative of the
mean within the corresponding cross-sectional box \rfig{CM-fluxcalc2}
also results in considerable uncertainties associated with the
volume-flux estimates. Because both current-meter moorings are located
near the center of the channel, away from the lateral boundaries, the
velocity estimates are most likely biased high. To derive a lower bound
for the transport the cross-sill velocities are assumed to decrease
linearly between the moorings and the lateral boundaries where they
vanish. Between the moorings the velocities are assumed to vary
linearly. The corresponding transport is calculated by partitioning the
cross-valley section \rfig{CM-fluxcalc2} into 6 boxes with the moorings
defining the lateral boundaries between the boxes. The velocities of
the two central boxes are set to the mean (projected) velocities of the
two instruments on either side and the velocities of the remaining 4
boxes are set to half the observed velocities of the single current
meters on their boundaries. The resulting volume-flux estimates are
listed in the third row of \rtabNP{CM-volflux}, labeled ``lower''.

The estimated range of the transport across \RS is
\mmmps*{40$--$100}{3} during the entire year, consistent with the
\mmmps*{65}{3} estimated from a hydraulic model \rsec{hydraulics}. It
is interesting to note that the ``best'' estimate is associated with
the smallest overall temporal variability (10\%).
