%======================================================================
%                    H Y D / F _ L O C A L . T E X 
%                    doc: Wed Apr  8 09:12:46 1998
%                    dlm: Sun Apr  2 21:40:08 2000
%                    (c) 1998 A.M. Thurnherr
%                    uE-Info: 135 51 NIL 0 0 72 3 2 4 ofnI
%======================================================================

\Subsection{sigma2}{Horizontal Density Distribution}

\begin{figure}[t]

\placeFig[1]{HYD/sig2200}

\caption[\F density and flow at \protect\m{2200}]{Mean horizontal
density distribution and instantaneous LADCP measurements (arrows)
between \m{2175} and \m{2225}; bathymetric contour interval is \m{250};
lightly shaded cells indicate below-average potential densities\REM{of
36.9368kgm\^[-3]}, darkly shaded cells are more than one standard
deviation ($\approx\!3\times10^{-3}$) denser than the
mean\REM{(36.9394kgm\^[-3])}, and intermediately shaded cells lie in
between.}

\labelFig{sig2200}

\end{figure}

\callout{\rFigNP{sig2200}} shows the horizontal distribution of
potential density at \m{2200}, i.e.\ at an intermediate depth between
the peak of \RR (\m{1950}) and the saddle of \RS (\m{2500}). The
shading of each cell is determined by the pressure-averaged \BGT
density measurements within its boundaries, with darker shades
indicating higher densities. At \m{2200} the SW basin water is
generally denser than the NE basin water. It is not clear if the
high-density water is confined to the northeastern part of the SW basin
because the lightly shaded cells found in the remainder of the basin
are all based on few data points from single \BGT lines.

\begin{figure}[t]

\placeFig{HYD/F_sigprof}

\caption[\F \RS density profiles]{Pressure-averaged density profiles
(above the sill depth of \RS) of the SW basin, \RS, and the NE basin;
the dotted line at \m{2000} indicate the depth below which the density
profiles diverge (see also \protect\rfigNP{F-Nprof}).}

\labelFig{F-sigprof}

\end{figure}

\begin{figure}[t]

\placeFig{HYD/F_Nprof}

\caption[\F \RS buoyancy frequency profiles]{Pressure-averaged buoyancy
frequency profiles (above the sill depth of \RS) of the SW basin and
the NE basin; the dotted lines is the same as in
\protect\rfigNP{F-sigprof}.}

\labelFig{F-Nprof}

\end{figure}

\callout{\rFigNP{F-sigprof}} shows the pressure-averaged potential
density profiles of the two deep basins and \RS. The horizontal density
difference between the basins has a maximum value of $4.5\times10^{-3}$
near \m{2250}. The different slopes of the density profiles between
\m{2000} and \m{2250} indicate that, on average, the isopycnals in this
depth range are more spread in the NE basin than in the SW basin. This
is confirmed by the corresponding buoyancy frequency profiles
\rfig{F-Nprof}. The stratification below \m{2400} is too weak to be
resolved accurately. An estimate of \ps{10^{-4}} is used in
\rsecNP{hydraulics} for the SW basin buoyancy frequency at \m{2500}.

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

\Subsection{LADCP}{Direct Current Observations}

\begin{figure}[t]

\placeFig{HYD/F_CM}

\caption[First week average current meter velocities]{Average current
meter velocities around \RR recorded during the first week of
deployment; the shading of the arrows is determined by the instrument
depth with black, gray, white indicating \m{2300}, \m{2100}, and
\m{1800}, respectively; bathymetric shading is the same as in
\rfigNP{AMAR-bridge}; contour interval is \m{250}.}

\labelFig{F-CM}

\end{figure}

In addition to the density distribution, \rfigNP{sig2200} also shows
the LADCP current measurements at \m{2200}. The two main features to
note are the consistent clockwise flow around the northern tip of \RR
(the individual measurements were taken over a period of three weeks)
and the strength of the currents on its western slope. \rFigNP{F-CM}
shows the averaged current meter velocities recorded during the first
week of deployment, i.e.\ immediately after the \F cruise. (The current
meter data are analyzed in detail in \rchapNP{CM}). On the sill the
velocities at \m{2300} (shown in black) are consistent in magnitude and
direction with the LADCP velocities shown in \rfigNP{sig2200} while the
strong boundary current on the western slope of \RR was not sampled.
Overall, the consistency between the two independent current
measurements is striking and suggests that the flow across \RS can be
considered quasi-steady. The current meters from mooring ``F''
\rfig{CM-stations} which is not shown in the figure indicate N- to
NE-ward mean flows of \mps{0.01$--$0.03} at all depths.

\begin{figure}[t]

\placeFig{HYD/altladcp15}

\caption[\F \RS LADCP profile]{Speed and density profiles in the
saddle of \RS.}

\labelFig{altladcp15}

\end{figure}

\begin{figure}[t]

\placeFig{HYD/meancur}

\caption[\F \RR boundary current]{Pressure-averaged flow velocity on
the western slope of \RR; each data point represents a \m{50} vertical
mean; the dotted line is the same as in \protect\rfigNP{F-sigprof}.}

\labelFig{meancur}

\end{figure}

\callout{\rFigNP{altladcp15}} shows a current speed profile from the
saddle of \RS. The layer of strong flow peaking at \mps{>\!0.25} has a
vertical scale of approximately \m{300}. Similar layers of intense
currents are apparent in four out of the seven profiles from the
western slope of \RR; they have vertical scales between \m{250} and
\m{400} and peak velocities between \mps{0.17} and
\mps{0.28}\REM{(\mps{0.24} mean)}. \callout{\rFigNP{meancur}} shows the
vertical structure of the \m{50}-averaged current field on the western
slope of \RR, derived from the seven LADCP profiles from the slope. The
layer of intense flow is confined to depths below \m{2000} where its
direction is much more uniform than above, indicating topographic
steering.

Layers of intensified flow near the seabed were observed in profiles
away from the western slope of \RR as well. They have typical vertical
scales of \m{\approx\!50} and peak velocities between \mps{0.1} and
\mps{0.15}. (It is interesting to note that a profile from the western
slope of a different topographic high (\lonW{33}{45}), \latN{36}{19} is
similar to the ones from the western slope of \RR, with a vertical
scale of \m{200} and a peak velocity of \mps{0.19}.)

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

\Subsection{hydraulics}{Flow Across \RS}

\begin{figure}[t]

\placeFig[1]{HYD/sigBGT07}

\caption[\F cross-sill density section]{Isopycnal contours from a
tow-yo across \RS; contour levels were chosen for uniform spacing with
depth; the shaded area indicates $\sigma_2>36.945$, found in the NE
basin only below \m{2600} except for a few profiles close to \RS.}

\labelFig{sigBGT07}

\end{figure}

\callout{\rFigNP{sigBGT07}} shows a selection of isopycnal contours
from a tow-yo across \RS (the black cross-sill tow of
\rfigNP{F-stations}). The contour levels were chosen so that their mean
depths are uniformly spaced. The isopycnal surfaces of the densest
water on the sill (at \km{14.5}) follow the topography descending into
the NE basin, consistent with northeastward cross-sill flow. The
associated spreading of the isopycnals below \m{2000} is indicative of
mixing with NE basin water. A second tow-yo across the sill which was
discarded because of instrument calibration problems \rsec{F-methods}
also shows a horizontal density gradient below \m{2000} caused by
down-sloping of the isopycnal surfaces in the NE basin.


The densest water observed on \RS \rfig{F-sigprof} has a (\BGT-derived)
\SigmaTwo value of $36.947$ which is idistinuguishable within
instrument intercalibration errors \rsec{F-methods} from the
(CTD-derived) potential density found within the deep basins on either
side of \RS \rfig{F-sigcompare}. This observation, together with the
virtually identical \ThetaS characteristics of the deep water of both
basins \rsec{sigma2}, indicates that \RS does not block the exchange of
deep water between the basins. The criterion for topographic blocking
to occur in linearly stratified uniform flows across two-dimensional
obstacles is that the dimensionless parameter
$\NhU=N_ih_b/U_i\gtrapprox2$\ls{NhU} for most obstacle shapes in
non-rotating flows \cite{review/baines87} and $\NhU\gtrapprox1.5$ for
rotating flows over a Gaussian ridge \cite{JAS/pierrehumbert+wyman85},
with $N_i$\ls{Ni}, $U_i$\ls{Ui} and $h_b$\ls{hb} denoting the upstream
buoyancy frequency, velocity and the height of the obstacle,
respectively. Using \ps{N_i=10^{-4}} \rsec{sigma2}, \mps{U_i=0.05}
(typical LADCP velocity away from \RR), and \m{h_b=700} (height of \RS
above the SW basin floor), $\NhU$ is approximately 1.4, i.e.\ close to
the limiting value. The uplifting of deep upstream water is driven by
the dynamic pressure reduction caused by the large flow velocities
across the sill; a phenomenon sometimes called Bernoulli Aspiration
\citeEG{inbook/kinder+bryden90}.

The horizontal density gradient across \RS is consistent with upstream
influence of an obstacle which has been observed for $\NhU$ as low as
$0.75$ \cite{JAS/pierrehumbert+wyman85}. The orientation of the
corresponding pressure gradient implies a mean overflow from the SW
into the NE basin. The observation that the density distributions of
the individual tows are consistent with the mean picture are consistent
with the view that the overflow across \RS can be considered
quasi-steady. The mutual consistency of the LADCP measurements around
\RR \rfig{sig2200} and the lack of indications for flow reversal also
support this view and furthermore indicate that tidal effects
\rsec{intwaves} are weak compared to the density-driven flow (see also
\rsecNP{CM-energy}). Further evidence for a quasi-steady flow field on
an even longer time scale is provided by the observation of a
persistent northward current on the western flank of \RR throughout an
ROV dive lasting 26h in August 1996 \cite{EOS/german+96} and by the
along-segment density gradient consistent with our data observed in
August--October 1992 \cite{JGR/wilson+95}.

An inviscid flow across a sill is (hydraulically) controlled if the
horizontal density distribution is asymmetric with respect to the sill
\cite{JFM/armi86}. \citeN{JPO/pratt86} shows how the inviscid
assumption of hydraulic models can be tested by evaluating the
non-dimensional parameter $P = C_d l/h$\ls{P} (relative importance of
hydraulic acceleration to deceleration caused by bottom friction) where
$C_d=10^{-3}$, $l$, and $h$ are the drag coefficient, the horizontal
distance over which the thickness of the active layer (see below)
changes significantly, and the mean thickness of this layer,
respectively. Conservatively estimating \km{l=15} and \m{h=300} from
\rfigNP{sigBGT07} results in an estimate of $P \approx 0.05$,
indicating that frictional effects are not important.
\citeN{DSR/killworth95} shows that the equivalence of hydraulic control
and flow maximization known from non-rotating hydraulics holds for
continuously stratified rotating flows controlled by sills and narrows
of arbitrary topography as well. Therefore, a hydraulic model can be
used to estimate the volume flux across \RS. The simple
\OneAndAHalfLayer reduced gravity model with zero upstream potential
vorticity introduced by \fullciteN{misc/whitehead+74} and extended to
uniform potential vorticity by \citeN{JFM/gill77} is considered
adequate for this purpose, mainly because it is the simplest model of
rotating hydraulics which has been tested in similar contexts
\cite{misc/whitehead97}. Without rotation it reduces to hydraulic flow
over a weir.

Layer models require estimates for the density differences between the
individual layers which are assumed to be homogeneous. The separation
of the water column into such layers is difficult except where there
are sharp temperature or salinity gradients coinciding with regions of
high vertical shear. In the \R profiles (e.g.\ \rfigNP{altladcp15})
there are no clear indications for such layers.
\citeN{GAFD/whitehead89} proposes a simple and consistent method for
dealing with this problem. It is based on the observation that in
hydraulically controlled stratified flows there is generally a well
defined \emph{bifurcation depth} below which the density profiles can
be separated into high density (upstream) and low density (downstream)
profiles. The bifurcation depth is taken to be the upstream depth of
the interface between the two model layers which makes this method
equivalent to defining the upstream interface depth as the depth of the
maximum density surface which remains horizontal over the sill
\citeEG{misc/mercier+bryden94}. The density difference between the
active lower and the passive upper layer is then set to the maximal
horizontal upstream-downstream density difference between the
bifurcation depth and the sill depth. (This method yields correct
values for homogeneous layers.) The respective values for the \R region
were estimated from \rfigNP{F-sigprof}; they are \m{2000} for the
bifurcation depth and $4.5\times10^{-3}$ for the density difference,
resulting in a reduced gravity estimate of \mpss*{g'\approx4.5}{-5}.
The vertical scale \m{h_u=500}\ls{hu} of the flow between the
bifurcation depth and the sill depth is consistent with the intensified
current layer shown in \rfigNP{meancur}.

In hydraulically controlled flows some suitably averaged velocity at
the control point is equal to a gravity wave speed so that the flow
becomes critical \citeEG{GAFD/whitehead89}. The only gravity waves
possible in a \OneAndAHalfLayer model are interfacial waves propagating
with a velocity of $(g'h)^{1/2}$. Because of the effect of rotation the
layer thickness is not constant even for a rectangular sill. Ignoring
this and setting $h=h_u$ the wave velocity estimate becomes \mps{0.14}
which is similar to the average velocity of \mps{0.17} between \m{2000}
and \m{2500} observed on \RS \rfig{altladcp15}.

The model of \fullciteN{misc/whitehead+74} is now used to estimate the
volume flux $Q$ and the width $w$\ls{w} of the current. The respective
expressions for a sill which is wide compared to the internal Rossby
radius of deformation are $Q=g'h_u^2/(2f)$ and $w=(2g'h_u)^{1/2}/f$. As
suggested by \citeN{GAFD/whitehead89}, we use the sill width at the
bifurcation depth for the comparison; the value of \km{7.5} taken from
bathymetric charts (e.g.\ \rfigNP{CM-fluxcalc2}) is much larger than
the Rossby radius \km{(g'h_u)^{1/2}/f\approx1.8} (\ps*{f=8}{-5} is the
Coriolis parameter at \degN{36}). The volume flux and current width
estimates become \mmmps*{Q\approx70}{3} and \km{w\approx2.7},
respectively. \citeN{GAFD/killworth94} derived parametrisations for
hydraulically controlled fluxes across sills of more realistic
geometries and upstream conditions and showed the expression of
\fullciteN{misc/whitehead+74} to be an upper bound. Using his
expression for parabolic sills leads to a reduction of the flux
estimate by 5\% which is an indication for the fact that \RS is wide
with respect to the internal Rossby radius. The Rossby number of the
flow $\Ro=U/fl\approx0.5$\ls{Ro} (based on the half-width of \RS
between the two deep basins (\km{l\approx\!5}) and a velocity scale of
\mps{0.2} estimated from \rfigNP{meancur}) indicates that the model
assumption of geostrophically adjusted flow across the sill is at least
partially violated. Effects of steady or time-variable (e.g.\ tidal)
barotropic forcing are ignored.

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

\REM{\subsection{Boundary Current on the Western Slope of \RR}

Based on the observations of the horizontal extent of the
light-scattering hydrothermal plume on the western slope of \RR
\rsec{P-1997} and the large along-slope currents encountered there
\rsec{LADCP} it appears likely that the northward flow takes the form
of a narrow boundary current hugging the contours of \RR. The
fundamental horizontal length scale $l$ for rotating flows is the Rossby
radius of deformation. Taking stratification into account $h/l=f/N$
defines an associated vertical length scale $h$ (sometimes called
\emph{deformation depth,} e.g.\
\citeN{JAS/pierrehumbert+wyman85}) as rotation tends to
constrict the horizontal scale of the flow while stratification
constricts its vertical scale. In boundary currents on a slope, on the
other hand, the ratio $h/l$ is prescribed externally by the slope
$\alpha$ of the topography.

\citeN{JPO/chapman+lentz97} present a model for the adjustment of an
along-isobar geostrophic boundary current with the slope on its right
in the northern hemisphere. As their model includes the effects of
stratification, rotation, and sloping topography it addresses the
problem outlined above. The model includes the effect of bottom
friction as well but, surprisingly, the scales of the resulting
currents are independent of the actual friction coefficients. The
thickness and along-slope velocity of the equilibrium current scale as
$h=(f/N)(F_0/f)^{1/2}$ and $U=N^2\alpha h/f$, respectively, where $F_0$
is the pre-adjustment inflow volume flux per unit depth. Noting that
$h$ does not depend on $\alpha$ the total volume flux $Q$ is set to
$F_0 h$ leading to $h=(fQ/N^2)^{1/3}$. Thus, the model imposes a
\emph{vertical} scale --- the horizontal scale is determined by the
slope, i.e.\ $l=h/\alpha$. Using
$Q=55\times10^3\mathrm{m}^3\mathrm{s}^{-1}$ estimated from the
hydraulic model \rsec{hydraulics} leads to a vertical scale estimate of
\m{h\approx260} which is consistent with the observations \rsec{LADCP}.
A typical value for $\alpha$ of 0.3 has been estimated from bathymetric
charts on the western slope of \RR. This leads to horizontal scale and
velocity estimates of \m{920} and \mps{0.25}, respectively.

When comparing these estimates to the observations it should be noted
that the model is based on the assumption of small Rossby number flow
and constant slope and buoyancy frequency. Furthermore, the
pre-adjustment inflow volume flux $F_0$ of the model has been replaced
by the volume flux estimate from the observations, i.e.\ by the fully
adjusted volume flux (the same method is used by
\citeN{JPO/chapman+lentz97} to compare the model results to a real flow
situation). The velocity estimate agrees well with the current
observations \rsec{LADCP} while the small horizontal length scale is
consistent with the general lack of observation of sloping isopycnals
below \m{2000} on the western slope of \RR (e.g.\
\rfigNP{sigBGT04}).}

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

\Subsection{leewaves}{The Topographic Wake}

The mean density profiles of the two deep basins \rfig{F-sigprof}
suggest that the hydraulically controlled flow is confined to depths
below \m{2000}. Between this level and the peak of \RR at \m{1950} the
incident current, of order \mps{0.05} \rfig{meancur}, flows over and
around the topography. Evaluating $\NhU$ \rsec{hydraulics} with
\ps{N_i=10^{-3}} \rfig{F-Nprof} and \m{h_b=50} results in a value of
$1.0$, indicating that blocking is unlikely. The characteristic
half-width $l_p$\ls{lp} of the top \m{50} of \RR is \km{1$--$2}.
Because \m{l_p\gg U_i/N_i=50} non-hydrostatic effects may be ignored,
while the corresponding Rossby number of $\approx\!0.5$ indicates that
rotational effects may not.

\begin{figure}[t]

\placeFig[1]{HYD/sigBGT04}

\caption[\F cross-ridge density section]{Isopycnal contours of a tow-yo
across \RR;  contour levels are the same as in \rfigNP{sigBGT07}.}

\labelFig{sigBGT04}

\end{figure}

The appropriate flow regime for these parameter values is the
\emph{rotating wave regime} of \citeN{misc/queney48} (see
\citeN{books/gill82}). In this regime, lee waves radiate energy (group
velocity) upwards and down-current at an angle between \deg{1} and
\deg{90} to the horizontal. The horizontal wavelength $\lambda$ of the
pattern of vertical stream line displacement close to a bell-shaped
ridge is $2\pi U_i/f$, which is \km{\approx\!4} using the \R parameters.
Because this is comparable to the width ($2\,l_p$) of \RR large-amplitude
lee waves are expected. \callout{\rFigNP{sigBGT04}} shows a selection
of isopycnal contours from a tow across \RR (the black cross-ridge tow
in \rfigNP{F-stations}). The pattern of isopycnal surface displacement
found in the lee of the ridge is consistent with the rotating wave
regime because it appears to be confined in the quadrant upwards and
downstream of the ridge peak and the wavelength directly over the peak
is \km{\approx\!5}. (The doming of the isopycnals in the eastern part
of the NE basin is not considered to be a topographic effect but is
consistent with tidal effects; see \rsecNP{intwaves}.) Two additional
\BGT tows across \RR show similar wave-like features although the
patterns there are less clear because of the decreased horizontal
resolution caused by larger towing velocities. These additional
observations together with the horizontal extent of the lee wave
patterns (\km{\approx\!10}) indicates that the topographic wake is most
likely caused by the mean flow (a velocity of order \mps{0.5} would be
required to propagate a disturbance over that distance in half a tidal
period).

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

\Subsection{intwaves}{Hydrographic Variability on Tidal Time Scales}

One of the main problems in interpreting hydrographic surveys like the
one presented here is the difficulty in separating spatial and temporal
scales, especially in the tidal range. (Based on re-sampling of the
same region over the four week data collection period, a measurable
shift of the hydrography over time scales of the order of weeks can be
excluded.) The vertical variability of the isopycnals shown in
\rfigNP{sigBGT04}, for example, is due to a combination of spatial as
well as temporal variability over the sampling time of \h{12}. If the
doming of the isopycnals in the NE basin were a spatial feature it
would imply anticyclonic flow around the basin, which is inconsistent
both with direct velocity measurements (e.g.\ \rfigNP{sig2200}) and
with density observations from four additional tows crossing the same
basin in different directions.

% - description of YOYO:
% 	- phase relationship with depth
% 	- amplitudes

\begin{figure}[t]

\placeFig{HYD/altyoyo}

\caption[\F CTD yoyo density]{Isopycnal contours of the \protect\h{10}
CTD yoyo; contour levels and shading are the same as in
\rfigNP{sigBGT07}.}

\labelFig{yoyo}

\end{figure}

To investigate the temporal variability of the hydrography on tidal
time scales a \h{10} CTD yoyo was done at a station close to \RS (the
black star with a white border in \rfigNP{F-stations}). Unfortunately,
the LADCP failed during the cast. \callout{\rFigNP{yoyo}} shows the
evolution of the isopycnal surfaces during sampling. Between the upper
turning point of \m{1600} and \m{\approx\!2250} the isopycnals show a
wave-like structure consistent with the semi-diurnal tidal period. The
phase of the isopycnal displacement appears to vary with depth. The
maximum amplitude near \m{1800} is \m{>\!100} but values of order
\m{50} are more typical. Close to the sea floor the structure is more
complex; the potential density inversions at \h{4.5}, \h{7}, and
\h{8.5} are not contouring artifacts. (There are indications from the
CTD altimeter for some horizontal displacement during the cast.)