%======================================================================
%                    C M / E N E R G Y . T E X 
%                    doc: Mon Mar 13 14:20:24 2000
%                    dlm: Wed Jun  7 22:46:03 2000
%                    (c) 2000 A.M. Thurnherr
%                    uE-Info: 403 17 NIL 0 0 72 3 2 4 ofnI
%======================================================================

\Subsection{CM-spectra}{Flow and Temperature Spectra}

% rationale & method

The daily-averaged overflow velocities shown in \rfigNP{CM-ovsticks}
are consistent with the conclusion of \rchapNP{HYD} that during the
1997 survey (i.e.\ on a time scale of weeks) the deep flow across \RS
was quasi-steady. Outside the core of the overflow and on longer time
scales a similar conclusion cannot be drawn, however. While averages
can be calculated from any time series, mean velocities are only
representative if the energetic time scales of the flow are sampled
adequately; this can be assessed by comparing the integral time scales
to the total length of the records \cite{JMR/flierl+mcwilliams77}. To
construct the lagged autocorrelations from which the integral time
scales are derived high frequency deterministic oscillations are
removed by filtering the data \citeEG{JPO/saunders87}. To determine the
filtering frequency ranges discrete power spectra (periodograms)
normalized to the mean squared amplitudes are calculated. Rotary
spectral methods \cite{DSR/mooers73,DSR/gonella72} are used for the
velocity data because \PVD observations (e.g.\
\rfigsNP{F-PVD-G}{F2-PVD-G}) and current meter animations indicate
preferentially \CW polarizations.

In the scope of this study no statistical analysis of the individual
spectral peaks is required, implying that simple methods can be used.
Because the flow energy at all frequencies is of interest the current
spectra are calculated from the original data without applying any
pre-conditioning (e.g.\ removing the mean, de-trending, pre-whitening,
etc.) while the mean is removed from the temperature records before
calculating the corresponding spectra. Tapering the data to decrease
end effects and \emph{leakage} as recommended by
\citeN{books/bendat+piersol71} was observed not to significantly change
the spectra of our data sets; without tapering the summed periodograms
are exactly equal to the mean squared amplitudes of the original data,
providing a convenient consistency check.

% CAVEAT required by Adam Schultz
%	- pre-conditioning
%	- sequence truncation

Because the data sequences are not de-trended before applying the
Fourier transforms the resulting power spectra are contaminated by
wrap-around discontinuities caused by the underlying periodicity
assumption. Removal of (linear or higher-order) trends can decrease
this problem in many cases but the effectiveness of de-trending depends
on the properties of the original data and on the parametric choice of
the trends to be removed. Because pre-conditioning methods such as
de-trending and pre-whitening are high-pass filters they distort the
low frequencies of the resulting spectra. Trends in the data (which are
removed by pre-conditioning) can be caused by low-frequency
oscillations which are insufficiently resolved, implying that integral
time scales calculated from de-trended data can underestimate the true
values and result in optimistic error estimates for the mean flow. In
the case of rotary spectra there is an additional problem concerning
the potential effects of pre-conditioning on the polarization
properties of the modified spectra. (See \citeN{books/hamming89} for a
discussion of additional problems which may arise when the data are
pre-conditioned by removing means and trends.) In summary, the problem
of how to pre-condition the data sequences before calculating power
spectra is a complex one and there are opposing considerations to be
taken into account. While it is acknowledged that the simple methods
used here are sub-optimal for the analysis of individual spectral peaks
the decision to use minimal pre-conditioning is based on the desire to
preserve the energy distribution and polarization properties
\rsec{CM-pies}, and the integral time scales \rsec{CM-asflow} derived
from the spectra.

A simple radix-2 FFT algorithm is used to estimate the power spectra.
The data sequences are shortened to the nearest power of two rather
than padded with zeroes \cite{books/hamming89}. The reduction in
spectral frequency resolution resulting from this truncation is not a
major concern because the energy distribution is primarily analyzed in
5 frequency bands which are much wider than the resolution bandwidth of
the individual spectra. 

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

\begin{figure}[t]

\REM{\placeTwoFigs[0.6]{CM/B23rotspec}{CM/B23tempspec}

\caption[Flow and temperature spectra]{Power spectra of the polarized
energy components (left panel) and temperature fluctuations (right
panel) measured by instrument B23 during passage of the warming event;
frequency and power are in units of cycles per day and squared
amplitudes, respectively; the vertical lines separate the frequency
into 5 bands, labeled above the plot (c.f.\
\protect\rtabNP{CM-fbands}); the arrows above the spectra indicate the
main peaks at tidal frequencies ($K_1$, $M_2$, $M_4$, $M_6$), a
tidal-inertial interaction frequency ($fM_2$, \citeNP{GRL/mihaly+98}),
and in the infra-diurnal band ($I$); $\omega^{-2}$ shows the GM72
roll-off in the internal-wave band (between $f$ and $N$ indicated below
the spectra); resolution bandwidth is \cpd*{9.4}{-2}; confidence
intervals are calculated with 16 degrees of freedom obtained from
frequency averaging \protect\cite{books/bendat+piersol71}.}

\labelFig{CM-spectra}}

\placeFig{CM/B23rotspec}

\caption[Rotary flow spectra]{Power spectra of the polarized energy
components recorded by instrument B23 during passage of the warming
event; frequency and power are in units of cycles per day and squared
flow speeds, respectively; the vertical lines separate the frequency
into 5 bands, labeled above the plots (c.f.\
\protect\rtabNP{CM-fbands}); the arrows above the spectra indicate the
main peaks at tidal frequencies ($K_1$, $M_2$, $M_4$, $M_6$), at a
tidal-inertial interaction frequency ($fM_2$, see
\citeNP{GRL/mihaly+98}), and in the infra-diurnal band ($I$);
$\omega^{-2}$ shows the roll-off of the GM72 model in the internal-wave
continuum (between $f$ and $N$, indicated below the spectra);
resolution bandwidth is \cpd*{9.4}{-2}; the confidence interval is
calculated with 16 degrees of freedom obtained from frequency averaging
\protect\cite{books/bendat+piersol71}.}

\labelFig{CM-rotspec}

\end{figure}

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

% spectra; discuss f (& K1), M4, M6

\rFigNP{CM-rotspec} shows the rotary flow spectra of the data recorded
by instrument B23 during passage of the warming event (c.f.\
\rfigNP{CM-ovsticks}) as an example. The energy is characterized by a
general decrease with increasing frequency with a number of
superimposed significant peaks, the strongest of which are labeled. In
addition to the tidal peaks at the $K_1$ and $M_2$ frequencies, there
is increased energy in a number of additional bands within the
internal-wave continuum between $f$ and $N$, indicated below the
spectra. ($N$ is taken from \rfigNP{F-Nprof}, i.e.\ it is the mean
buoyancy frequency of the SW basin during the 1997 hydrographic survey
which may not accurately represent the upper frequency limit of the
internal wave band at mooring B during passage of the warming event.)
The peak at the sum of the inertial and the semi-diurnal tidal
frequencies (labeled $fM_2$) is indicative of non-linear interaction
between the two bands \cite{GRL/mihaly+98}. The $M_4$ peak, normally
only found in shallow water, was recently observed near a mid-ocean
ridge in the northeast Pacific \emph{(ibid).} In some of the spectra
the $M_4$ and the $fM_2$ peaks are as high as the peaks in the diurnal
band. In the pre-warming (and in most of the post-warming) spectra there
are no strong near-inertial peaks at \m{2300}, but at \m{1800} and
\m{2100} and in nearly all records during passage of the warming event
they are significant at the 5\% level. (Because the resolution
bandwidth is of the order of the separation between $K_1$ and $f$ the
respective contributions are not always separated by spectral gaps.)

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

\placeFig{CM/B23tempspec}

\caption[Temperature-amplitude spectrum]{Power spectrum of the
temperature fluctuations measured by instrument B23 during passage of
the warming event; power is in units of squared temperature;
the annotations are the same as in \protect\rfigNP{CM-rotspec}.}

\labelFig{CM-tempspec}

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

\rFigNP{CM-tempspec} shows the spectrum of the temperature fluctuations
corresponding to the velocity data of \rfigNP{CM-rotspec}. Most of the
energy is contained in the low-frequency band, consistent with the
\degC{0.1} warming observed during that part of the record (c.f.\
\rfigNP{CM-ovsticks}).\REM{The pre/post temperature spectra have approx
an order of mag less energy at the lowest frequencies.} The significant
peaks in the temperature spectrum have corresponding signatures in the
flow spectral-energy distribution, especially at the main tidal
frequencies and in the near-inertial band. The variability of the
spectral-peak magnitudes from different instruments is high, however
--- in simultaneous spectra at the same depth derived from other
instruments, the peak power of the temperature fluctuations in the
semi-diurnal tidal band is more than an order of magnitude
larger\REM{G23,C23} than the one shown without a corresponding increase
in the flow spectral energy. This could be either because of more
baroclinic structure of the flow \citeEG{JGR/mueller+78} or because of
higher temperature stratification.

Comparison of the flow and temperature spectra reveals that some of the
prominent spectral peaks in the former have only weak or no signatures
in the latter. A notable case in point is the peak labeled $I$ at
\cpd{0.23}. In the flow spectra plotted against a logarithmic frequency
axis it appears as a broad and possibly only marginally significant
feature. Its bandwidth of \cpd{0.3} is in fact similar to that of the
$M_2$ and $K_1$ peaks. This and similar oscillations are much easier to
detect in autocorrelation plots (e.g.\ \rfigNP{CM-autocorr}) than in
power spectra.

The autocorrelations show a range of oscillations with periods greater
than a day, varying in magnitude and frequency both in time and space.
On \RS (moorings B and D) periodic motions with frequencies of
\cpd{0.2$--$0.3} are observed at \m{2300} (and sometimes also at
\m{2100} but much less clearly so) during all three temporal regimes
but they are much weaker after passage of the warming event. (Close
inspection of \rfigNP{CM-ovsticks} also shows a signature of this
oscillation at \m{2300} where the daily-averaged velocities are
``bunched together'' in groups of 4--6.) Near the high-frequency limit
of the sub-diurnal band there is a well-defined oscillation with a
frequency of \cpd{0.8} on the eastern flank of \RR (mooring C)
increasing with depth during both the pre-warming and the warming
regimes and a similar one with a frequency of \cpd{0.5} after passage
of the event. The regular oscillation with the lowest observed
frequency (corresponding to a period of \d{14}) is manifest in the
records from instruments C18, D18 and F21 during passage of the event.
There does not appear to be any preferred polarization of the energy of
these oscillations.

Given the enclosed nature of the rift valley below \m{2000} it is
assumed that at least some of the oscillations with periods greater
than a day are free basin modes. Because of the complexity of the
patterns this is difficult to confirm, however. Using a range of values
for the effective depth of the rift valley and estimating corresponding
reduced gravities from \rfigNP{F-sigcompare}, the Rossby radius of
deformation becomes \km{1$--$7}, i.e.\ it is smaller than the width of
the rift valley at the depth of the current meters, indicating that the
effects of rotation cannot be ignored. For large rotation rates, i.e.\
$f\gg\omega$ where $\omega$\ls{omega} is the frequency of the observed
oscillations, the free modes of enclosed rectangular or circular
unstratified basins with vertical walls are similar to Kelvin waves
propagating around the rims of the basins
\citeEG{books/leblond+mysak78}\REM{p.295}. With the parameters used to
estimate the Rossby radius the gravity wave speed\REM{Kelvin waves move
along the coast with the long non-rotating gravity wave speed, Gill
1982, p.378} becomes \mps{0.1$--$0.5} which, together with periods
between 1 and 14 days, yields a range of horizontal scales of
approximately \km{10$--$630}. If the fastest (slowest) waves are
assumed to be associated with the shortest (longest) periods this range
reduces to \km{45$--$120} which is consistent with the horizontal
extent of the \A segments.

Another type of wave which can potentially account for some of the
oscillations are short (compared to the Rossby radius of deformation)
topographic planetary (or Rossby-) waves, possibly generated by the
vorticity perturbation introduced into the water column by the
hydrothermal plume. (Long topographic Rossby waves are barotropic which
is inconsistent with the observations, whereas the short waves decay
exponentially with distance from the sloping bottom.) The dispersion
relation of short topographic Rossby waves in uniformly stratified
basins with uniformly sloping bottom boundaries is
 %
\begin{equation}
 %
\omega = -\gamma N \frac{k_l}{(k_l^2 + k_x^2)^{1/2}}
\label{eqn:disptopo}
 %
\end{equation}
 %
\citeEG{books/leblond+mysak78}, where $\gamma$\ls{gamma} is the bottom
slope, and $k_l$\ls{kl} and $k_x$\ls{kx} are the along-slope and the
cross-slope wave numbers, respectively. The negative sign indicates
that $k_l$ is negative, i.e.\ that the waves propagate in the
``topographic westward'' direction (with the slope shallowing on their
right). Assuming $k_l$ and $k_x$ to be of similar size, taking a value
of \ps*{N=5}{-4} from \rfigNP{F-Nprof} at \m{2300}, and estimating the
representative slope of the rift valley at that depth to be
approximately 0.1--0.2 (e.g.\ \rfigNP{CM-fluxcalc2}) yields a frequency
range of \cpd{0.7$--$1.4}, indicating that short topographic Rossby
waves can account for some of the observed oscillations.

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

\Subsection{CM-pies}{Bulk Energy Distribution}

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

\begin{table}[t]

\begin{center}\begin{tabular}{|c|rrl|}\hline
%----------------------------------------------------------------------
Band & Frequency 			& Period 		& Significance \\ \hline\hline
%======================================================================
LF	& \cpd{0$--$0.143}  	& \w[]{\geq\!1} & low frequencies \\
IF	& \cpd{0.143$--$0.6}	& \h{40}--\w[]{1} & free basin modes? topographic Rossby waves? \\ 
DF	& \cpd{0.6$--$1.44}		& \h{16.7$--$40}& diurnal band (including near-inertial range) \\
SD	& \cpd{1.44$--$2.4}		& \h{10$--$16.7}& semi-diurnal band \\
HF	& \cpd{2.4$--$12}		& \h{2$--$10}	& high frequencies \\ \hline
%----------------------------------------------------------------------
\end{tabular}\end{center}

\caption[Current-meter frequency bands]{Current-meter frequency bands.}

\labelTab{CM-fbands}

\end{table}

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

\begin{figure}[t]

\placeFig[\fullwidth]{CM/pies}

\caption[Flow-energy partition]{Pre-warming energy partition of the
current-meter records of moorings B and F; the limits of the frequency
bands are listed in \protect\rtabNP{CM-fbands}; pie areas are
proportional to the energies which are also listed in units of
\mmpss{10^{-4}} above the pies; shading is determined by the relative
polarizations with shaded sectors denoting the \ACW components.}

\labelFig{CM-pies}

\end{figure}

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

% separation into energy bands

The primary goal of the current-meter analysis is an assessment of the
variability of the along-valley flow. Inspection of the individual
spectra, such as the ones shown in \rfigNP{CM-rotspec}, suggests that
the flow energy is dominated by contributions from a small number of
frequencies, most notably within the low-frequency, the diurnal and the
semi-diurnal tidal bands. Because the relative contributions from
different frequencies are difficult to compare on log-log plots, the
resolved frequency range is subdivided into 5 bands
(\rfigsNP{CM-rotspec}{CM-tempspec}, and \rtabNP{CM-fbands}). The
boundaries between the frequency bands were chosen to coincide with
spectral gaps in most records. The diurnal band contains both tidal
frequencies and the near-inertial range. The limits of the IF band were
selected to encompass the more energetic of the oscillations with
periods greater than a day \rsec{CM-spectra}. \rFigNP{CM-pies} shows
the energy partition of the pre-warming velocities recorded on moorings
B and F. Mooring F was selected because it represents the only record
for the continuation of the along-valley flow \km{30} NE of \RS
\rfig{CM-stations}.

In the pre-warming records the low-frequency components dominate in the
overflow across \RS (B23, D23), at \m{2300} on mooring F, as well as at
intermediate depths on the western slope of \RR (G21) and at the
southernmost mooring in the NE basin (H21). While this pattern changed
in a complex way during passage of the warming event, the post-warming
energy distribution is similar to the pre-warming state.\REM{A notable
exception being the deep NE basin mooring E which changed from strongly
tidally dominated before the warming to weakly low-frequency dominated
during and after the event.} Within the core of the overflow the
pre-warming low-frequency component is nearly an order of magnitude
larger than the tidal flow. The oscillations in the IF band are not
very energetic in the pre-warming records; during passage of the
warming event their contributions become as high as half that of the
low-frequency component in the \RS overflow, however\REM{B23} (e.g.\
\rfigNP{CM-rotspec}).

The only current meter mooring deployed on the western slope of \RR (G)
is also the one closest to (\km{1.5} north of) the hydrothermal vent
field. The flow observations at mooring G indicate a baroclinic
structure of the tidal oscillations\REM{both in the SD and the DF
bands} with the lowest energy at \m{2100}, the mean depth of the
hydrothermal particle plume \rchap{plume}. A direct influence of the
plume on the dynamics on the western slope of \RR cannot be excluded;
LADCP observations \rsec{LADCP}, which can easily be correlated with
hydrothermal particle signals, do not show any dynamical plume
signatures however and, given the vertical range of the near-source
plume observations of the 1997 survey \rsec{plumeprofs} as well as the
corresponding horizontal variability \rsec{plumeplan}, it seems
doubtful that there should be a strong signature at \m{2100} close to
the vent field. Topographic effects are therefore considered more
likely explanations for the flow recorded on mooring G.

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

\begin{center}\begin{tabular}{|c||c|c|c|}\hline
%----------------------------------------------------------------------
Depth	&	Pre-Warming	& Warming	& Post-Warming \\ \hline\hline
%======================================================================
\m{1800} &	0.60 		& 0.45 		& 0.57 \\
\m{2100} &	0.35 		& 0.28 		& 0.35 \\
\m{2300} &	0.13 		& 0.24 		& 0.20 \\ \hline
%----------------------------------------------------------------------
\end{tabular}\end{center}

\caption[Mean rotary coefficients]{Depth-averaged rotary coefficients
$C_r$ (see text for definition) of the flow observed during the three
temporal regimes.}

\labelTab{CM-Cr}

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

The energy polarization observed at all moorings resembles that shown
in \rfigNP{CM-pies} with the dominance of the \CW polarization
decreasing with increasing depth and with decreasing frequency.
Preferentially CW-polarized rotary currents at frequencies above
approximately \cpd{1} have been observed near mid-ocean ridges
elsewhere \citeEG{GRL/mihaly+98}. The decreasing polarization with
increasing depth observed in our data is consistent with topographic
steering. The mean flow (and also the energy at the Nyquist frequency)
is unpolarized by definition. \rTabNP{CM-Cr} shows the mean
polarizations during the three temporal regimes in terms of
depth-averaged rotary coefficients of the entire spectra $C_r =
(S_--S_+) / (S_-+S_+),$\ls{Cr} where $S_-$\ls{Sm} and $S_+$\ls{Sp} are
the \CW- and the \ACW-polarized rotary spectral components,
respectively \cite{DSR/gonella72}. Consistent with the observations
from individual moorings the mean rotary coefficients decrease with
increasing depth. During passage of the warming event the \CW dominance
decreased at \m{1800} and at \m{2100} while it increased at \m{2300}.
(The rotary coefficients at \m{2300} are dominated by the low-frequency
component of the \RS overflow.)

