%======================================================================
%                    H Y D / F _ L O C A L . T E X 
%                    doc: Wed Apr  8 09:12:46 1998
%                    dlm: Wed Jun  7 22:13:41 2000
%                    (c) 1998 A.M. Thurnherr
%                    uE-Info: 430 13 NIL 0 0 72 3 2 4 ofnI
%======================================================================

\Subsubsection{sigma2}{Horizontal Density Distribution}

\begin{figure}[t]

\placeFig[1]{HYD/sig2200}

\caption[1997 density and flow at \protect\m{2200}]{Horizontal density
distribution and instantaneous LADCP velocity 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 derived from few data points from single \BGT lines.

\begin{figure}[t]

\placeFig{HYD/F_sigprof}

\caption[1997 \RS density profiles]{Pressure-averaged potential density
profiles of the SW basin, \RS, and the NE basin; the dotted line at
\m{2000} indicates 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[1997 \RS buoyancy frequency profiles]{Pressure-averaged
buoyancy frequency profiles of the SW and the NE basins; the dotted
line 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 below for the
SW basin buoyancy frequency at \m{2500}.

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

\Subsubsection{LADCP}{Direct Flow Observations}

\begin{figure}[t]

\placeFig{HYD/F_CM}

\caption[1997 current-meter velocities (first week)]{Average
current-meter velocities in the vicinity of \RR recorded during the
first week of mooring deployment; shading of the arrows is determined
by the instrument depths with black, gray and 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 large velocities observed on its western slope. \rFigNP{F-CM}
shows the current meter velocities averaged over the first week of
deployment, i.e.\ immediately after the \F cruise. (The current-meter
data are analyzed in \rchapNP{CM}). On the sill the velocities at
\m{2300} (black arrows) 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 velocity measurements is
striking and suggests that the flow across \RS can be considered
quasi-steady over periods of weeks. The current meters from mooring
``F'', which is not shown in the figure, indicate N- to NE-ward mean
flow of \mps{0.01$--$0.03} between \m{1800} and \m{2300}.

\begin{figure}[t]

\placeFig{HYD/altladcp15}

\caption[1997 \RS flow and density profiles]{Flow speed and density
profiles from the saddle of \RS.}

\labelFig{altladcp15}

\end{figure}

\begin{figure}[t]

\placeFig{HYD/meancur}

\caption[1997 \RR boundary current]{Pressure-averaged flow field 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 flow-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 flow speeds 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 velocity field on the western
slope of \RR derived from all seven LADCP profiles. The layer of
intense flow is confined to depths below \m{2000} where the 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 flow speeds 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 (near
\lonW{33}{45}/\latN{36}{19}) is similar to the ones from the western
slope of \RR, with a \m{200}-thick layer of intensified flow peaking at
\mps{0.19}.)

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

\Subsubsection{hydraulics}{Flow Across \RS}

\begin{figure}[t]

\placeFig[1]{HYD/sigBGT07}

\caption[1997 cross-sill density section]{Isopycnal contours from a
tow-yo across \RSp{}{;} 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; total density range is 36.874--36.949
(c.f.\ \rfigNP{F-sigprof}). The isopycnal surfaces of the densest water
on the sill follow the topography descending into the NE basin,
consistent with NE-ward cross-sill flow. Downstream of the sill saddle
the isopycnal surfaces below \m{2000} exhibit vertical spreading
consistent with \rfigNP{F-sigprof}. This effect may indicate high rates
of diapycnal mixing, similar to observations in cross-sill flows
elsewhere \citeEG{Nature/polzin+96}. A second cross-sill tow-yo 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
\BGTp{(}{}-derived) \SigmaTwo value of $36.947$ which is
indistinguishable within the instrument intercalibration uncertainties
\rsec{F-methods} from the (CTD-derived) potential density found within
the deep basins on either side of the sill \rfig{F-sigcompare}. This
observation, together with the virtually identical \ThetaS
characteristics of the deep water in both basins \rsec{F-waterorigin},
indicates that \RS does not block the along-valley flow. The criterion
(derived from laboratory and numerical experiments) for topographic
blocking to occur in linearly stratified uniform flows across
two-dimensional obstacles is that the dimensionless parameter
(sometimes termed a Froude number) $\NhU=N_ih_b/U_i\gtrapprox2$\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}. $N_i$\ls{Ni}, $U_i$\ls{Ui} and
$h_b$\ls{hb} denote the upstream buoyancy frequency, flow speed and the
height of the obstacle, respectively. Using \ps{N_i=10^{-4}},
\mps{U_i=0.05} (typical LADCP-measured speed away from \RR), and
\m{h_b=700} (height of \RS above the SW basin floor), $\NhU$ becomes
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 values of $\NhU$
as low as $0.75$ \cite{JAS/pierrehumbert+wyman85}. The orientation of
the corresponding pressure gradient implies a mean flow from the SW
into the NE basin. The observation that the density distributions of
the individual tows (e.g.\ \rfigNP{sigBGT07}) are consistent with the
mean picture \rfig{F-sigprof} suggests that the overflow across \RS can
be considered quasi-steady on the survey time scale (\d{24}). Evidence
for persistent flow on even longer time scales is provided by the
current meter measurements \rchap{CM}, by the observation of a
northward current on the western slope of \RR throughout an ROV dive
lasting 26h in August 1996 \cite{EOS/german+96}, and by the
along-segment hydrographic gradients, consistent with our data,
observed between August and October 1992 \cite{JGR/wilson+95}.

In the case of an inviscid steady two-dimensional flow along a channel
of slowly varying geometry a horizontally asymmetric density
distribution (i.e.\ differing reservoir conditions) implies that the
flow is (hydraulically) controlled \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} (indicating the relative importance of hydraulic
acceleration and deceleration caused by bottom friction) where
$C_d(=10^{-3})$\ls{Cd}, $l$, and $h$ are the drag coefficient, the
horizontal distance over which the thickness of the active lower layer
(see below) changes significantly, and the mean thickness of that
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 can be ignored.
\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 (c.f.\ \rchapNP{BOF}) with zero
upstream potential vorticity introduced by
\fullciteN{misc/whitehead+74} is considered adequate for this purpose,
mainly because it is the simplest model of rotating hydraulics which
has been tested in the oceanographic context \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 vertical hydrographic gradients coinciding with regions of
high shear. In the \F 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 a hydraulically controlled stratified
unidirectional flow the mean upstream/downstream density difference is
generally confined below a well-defined depth (indicated by the
horizontal lines in \rfigsNP{F-sigprof}{F-Nprof}). This depth is termed
the \emph{bifurcation depth} and is taken to be the upstream level of
the interface between the two model layers, which makes this method
equivalent to defining the interface as 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 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 flow across \RS are estimated from
\rfigsNP{F-sigprof}{F-Nprof}; 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{\hu=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 is equal to a gravity wave speed so that the flow becomes
critical \citeEG{JFM/armi86}. The only gravity waves possible in a
\OneAndAHalfLayer model are interfacial waves propagating with a speed
of $(g'h)^{1/2}$. Defining the appropriate lower-layer depth $h$ in a
rotating system is not straightforward because it is not constant even
across a flat-bottomed channel. However, if it is assumed that $\hu$ is
a representative depth, the wave speed becomes \mps{0.15} which is
similar to the average flow speed 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 overflow current. The
respective expressions for a sill which is wide compared to the
internal Rossby radius of deformation are $Q=g'\hu^2/(2f)$ and
$w=(2g'\hu)^{1/2}/f$. As suggested by \citeN{GAFD/whitehead89}, we use
the sill width at the bifurcation depth for the comparison with the
Rossby radius; the value of \km{7.5} estimated from bathymetric charts
(e.g.\ \rfigNP{CM-fluxcalc2}) is much larger than the Rossby radius
\km{(g'\hu)^{1/2}/f\approx1.7} (where \ps*{f=8.6}{-5} is the Coriolis
parameter at \degN{36}). The volume-flux and current-width estimates
become \mmmps*{Q\approx65}{3} and \km{w\approx2.5}, respectively.
\citeN{GAFD/killworth94} derived parameterizations 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 consistent with the observation that \RS is
wide compared to the 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{\subsubsection{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{F-plume} 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
flow speed 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 flow speed 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}).}

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

\Subsubsection{leewaves}{A 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 $\NhU$ (see above) with
\ps{N_i=10^{-3}} \rfig{F-Nprof} and \m{h_b=50} results in a value of
$1.0$, indicating that no blocking is expected. The characteristic
half-width $l$ of the top \m{50} of \RR is \km{1$--$2}. Because \m{l\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 are important.

\begin{figure}[t]

\placeFig[1]{HYD/sigBGT04}

\caption[1997 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 parameters is the \emph{rotating
wave regime} of \citeN{misc/queney48} (see \citeNP{books/gill82}). In
this regime, lee waves radiate energy (group velocity) upwards and
down-stream 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 becomes \km{\approx\!4} using the \R parameters. Because this is
comparable to the width ($2\,l$) 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 of
\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 because 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-wave
pattern; it is consistent with tidal effects, however; see below.) Two
additional \BGT tows across \RR show similar wave-like features
although the patterns there are less clear because of decreased
horizontal resolutions caused by larger towing velocities. These
observations together with the horizontal extent of the lee-wave
patterns (\km{\approx\!10}) indicate 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).

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

\Subsubsection{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 \d{24} of data collection, a measurable shift of
the hydrography over this time scale can be excluded.) The vertical
displacement of the isopycnals shown in \rfigNP{sigBGT04}, for example,
is caused by a combination of spatial and 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[1997 CTD-yoyo density observations]{Isopycnal contours of the
\protect\h{10} CTD yoyo; contour levels and shading are the same as in
\rfigNP{sigBGT07}; total density range is 36.845--36.948.}

\labelFig{yoyo}

\end{figure}

To investigate the temporal variability of the hydrography on tidal
time scales a \h{10} CTD yoyo (12 individual down-/up-casts) between
\m{1600} and the sea bed at \m{2400} was done at a station close to \RS
(the black star with a white border of \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 at \m{1600} and
\m{\approx\!2250} the isopycnals show a wave-like structure consistent
with the semi-diurnal tidal period. The phase of isopycnal displacement
appears to vary with depth. The maximum amplitude near \m{1800} is
\m{>\!100} (similar to the observations of \citeN{EPSL/rudnicki+94} at
the TAG hydrothermal site) 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.)

