% Scott's thesis: Chapter 4 
 
\chapter{Plumes and Heat Flux Above the Ridge -- Multidirectional Flow}
\label{ch:upper}

Above the Endeavour ridge, multidirectional mean flow and amplified
oscillations constitute the currents near the equilibration depths of
hydrothermal plumes from high $B$ sources.  Accurate quantification of heat
flux in this type of flow is operationally more difficult than in the axial
valley where the flow is rectified and dominated by the semi-diurnal tide
(Chapter~\ref{ch:lower}).  Fundamentally, 4 side surfaces must be accounted for
instead of 2, meaning that there is approximately twice the observational
demand during any given oscillation.  Faster speeds, shifting directions, and
amplified oscillations at multiple frequencies conspire to generate more
variable plume distributions and heat fluxes.

During investigations of equilibrated plumes above ridge crests, plume
distributions are often described as more variable or unpredictable than
expected.  The assumption of a unidirectional long-term mean flow and an
anticipated downstream direction complicated the experiment of
\citeNP{rosenberg+88}; the Endeavour plume distribution was judged steady
enough to assess the standing stock of radon only during $\sim$5 of the
23\,days devoted to the assessment.  During the
\href{http://bromide.ocean.washington.edu/zephyr/}{1995 Mixing Zephyrs field
program (bromide.ocean.washington.edu/zephyr)} a regional hydrographic survey
of 48~VOTs above the Endeavour segment revealed a plume distribution that was
generally centered over the MEF, but extended in different directions on
different days \cite{veirs+99}.  While the plume was often elongated to the southwest, it was
intermittently absent in the southwest quadrant and present in the southeast or
northeast quadrants.  Equilibrated hydrothermal plumes with laterally
continuous distributions on scales of $\sim$10\,km have been observed
$\sim$5\,km from known sources in rotary flow on the southern Juan de Fuca
ridge \cite{baker+massoth86}, but over the Endeavour segment the plume
distribution commonly changes on weekly \cite{kadko+90} to hourly time scales.
As will become evident in this chapter, plume variability and inhomogeneity
appears to be the norm above the ridge crests within a few km of the MEF.  The
``patchiness'' of the Endeavour plume \cite[,~page~203]{franks92} has
complicated the analysis and interpretation of results from studies of the
downstream evolution of plume properties (\emph{e.g.}~\citeNP{mclaughlin98}).

Efforts to measure the heat flux from hydrothermal fields or ridge segments by
intercepting equilibrated plumes have historically generated results with large
formal uncertainties (\emph{e.g.}~\citeNP{baker+massoth87},
\citeNP{thomson+92}).  While some sources of uncertainty are peculiar to the
methods of each study, I posit in this chapter that the pooling and streaming
of plumes in multidirectional flow generates a level of hydrographic
variability that is commonly observed over Endeavour ridge and is probably
responsible for the wide-ranging estimates of MEF heat flux.  
%Both fast- and slow-spreading centers are likely to have combinations of
%topography and flow that require the approaches presented in this chapter. 

This chapter begins with a presentation of hydrographic observations made above
the ridge crests during the Flow Mow study, focusing on spatial and temporal
variability of plumes.  With a subset of the observations, I estimate the net
horizontal heat flux through the upper control volume (Figure~\ref{fmcv}).  I
then use a puff model to study how plumes disperse in multidirectional flow
above the ridge.  The model results guide the interpretation of the
hydrographic observations and also indicate how measured heat flux precision
increases with longer observation periods.  Finally, I compare the observed net
horizontal heat flux through the control surfaces of the upper Flow Mow control
volume with the vertical flux into the volume measured by \citeN{stahr+03}.
This leads to a discussion of sources of uncertainty in horizontal fluxes
estimated above the ridge in this chapter, as well as in previous Endeavour
endeavors. 

%% Might want to insert a bit more theory here (riseheight.tex), or perhaps in discussion?

\section{Observations above the ridge crests}

While ABE operated exclusively below $\sim$2100~m depth throughout the Flow Mow
study, the majority of the CTD casts extended from within a few meters of the
sea floor up to $\sim$1800~m, well above the maximum rise height for typical
MEF plumes.  The upper 300~m of these casts provide insights into the
distribution and dynamics of Endeavour plumes that have risen out of the axial
valley.  As mentioned in the previous chapter, these upper-level plumes either
originate directly from high $B$ sources, or represent plumes from low $B$
sources that have been entrained into higher-rising plumes.

In this section, I characterize the spatial and temporal variability of plumes
near the upper Flow Mow control volume using 3 basic types of CTD stations.
First, I present time series from the NoMEF and SoMEF areas
(Figure~\ref{abebathy}), including VOTs and VOCs from 1800--2200\,m.  Then I
describe VOTs along the 
%north, south, and east 
north and south side surfaces of the control volume, also from 1800--2200\,m.
%Remarkably, the west side surface was characterized only during a single ABE
%dive, and on that survey coverage was partial.  
The last suite of observations is a series of 5 circumnavigations of the
control volume perimeter accomplished with a slow ($\sim$0.1\,m/s ship speed)
VOT from 1850--2070\,m.  This $\sim$24\,hr ``ribbon'' of data around the MEF,
acquired despite being particularly mind-numbing for the winch operators and
mates, resolves changes in plume distribution on hourly time scales and 50\,m
lateral scales.

%The Flow Mow hydrographic survey was conducted at an exceptionally fine scale
%and with unusual proximity to well-mapped hydrothermal sources.  
%
% Some of the plumes were likely still buoyant, but I haven't checked that...!

\subsection{Time series north and south of the MEF: 1800--2200\,m}

It was particularly exciting to collect the long time series from the NoMEF and
SoMEF areas $\sim$500\,m north and south of the MEF (Figure~\ref{abebathy},  
\emph{c.f.}~Section~\ref{sec:mef_dth_lower})
because even at sea it was apparent that the hydrographic variability was
surprisingly high and had some tidal periodicity.  Figure~\ref{full_waterfall}
shows the $\Delta\theta$ and backscatter data from all CTD stations in the 2
areas, over the full depth range of 1800--2200\,m.  The most prominent features
of the hydrographic evolution are the sharp upper boundary, the relatively
warm, particle-rich waters centered near 2000$\pm$100\,m, and the underlying
cooler, clearer fluid within $\sim$100\,m of the valley floor.  

\begin{sidewaysfigure} \begin{center}
\includegraphics*[width=.45\textwidth]{figs/upper/Dth.nosomef.image.all2.eps}
\hfill
\includegraphics*[width=.45\textwidth]{figs/upper/bks.nosomef.image.all2.eps}
\caption[Time series of backscatter and $\Delta\theta$ north and south of MEF
(1800--2200\,m)]{Time series of spatially and temporally averaged
$\Delta\theta$ (left) and backscatter (right) from CTD surveys of the NoMEF
(upper) and SoMEF (lower) areas.  Dotted black line is CTD track.  Averaging
bins are $1/2$\,hr wide and 5~m high.  Vertical black lines mark temporal breaks
($\sim$1-5~days) between stations, each of which is numbered in lower left
corner.  White areas have no data (or $\Delta\theta<0^\circ$C).  See
Figure~\ref{nosomef_waterfall} for additional notes and $\Delta\theta$ detail
in bottom 100~m.} \label{full_waterfall} \end{center} \end{sidewaysfigure}

The upper boundary of the plume, defined by steep vertical gradients in
$\Delta\theta$ and backscatter, occupies a mean depth of $\sim$1890\,m, both
north and south of the MEF.  The mean depth varies temporally, however, with an
amplitude of $\sim$25\,m.  During the long stations (28 and 33) the boundary
undulates vertically with an approximate period of 12\,hr, implying that the
depth of the upper boundary is related to the semi-diurnal tide.  The currents
recorded at the uppermost meter on the southern mooring (FM-S250, 1942\,m,
Table~\ref{cm_table}) are indeed dominated by semi-diurnal oscillations during
both station 28 and 33, but there is no obvious correlation of boundary height
and hourly mean velocity.  At station 28, for example, the upper boundary
reaches high points at 2.5 and 8\,hr elapsed time, when the flow at the FM-S250
was 5\,cm/s southeast and 10\,cm/s southwest, respectively; at the intermediate
low near 4.5\,hr, the flow was 7\,cm/s southwest.  
%The cross flow within the axial valley could also have affected the maximum
%rise height, but the record from the near-bottom meter on the same mooring
%(FM-S50, 2142\,m) shows a similar evolution in speeds, though in distinct
%directions.  
While plume theory suggests that rise height will be reduced in faster cross
flows \cite{middleton+thomson86}, the maximum and minimum elevations of the
upper boundary occurred at similar mean flow speeds.  Of course, a
straightforward correlation should not be expected because the 500\,m lateral
distance between known sources and the monitoring stations could result in lags
of order $\sim$(500\,m)/(0.05\,m/s)$\sim10^4$\,s, or about 3\,hr.  An
additional reason to not expect direct correlations is that the flow history
can juxtapose plumes that experienced different conditions while reaching
equilibrium.  

% Not a bad place to talk about how much heat flux change is needed to change
% rise height by 50m...

The temporal evolution of the warm, particle-rich plume centered near 2000\,m
is a manifestation at NoMEF of vent fluid advected from either the MEF or High
Rise fields and background water brought in from off-axis.  The
\href{http://econscience.org/scott/pubs/thesis/animations.html}{puff model}
with flow determined by the record from FM-S250 simulates the combined effects
of flow history and lateral separation of hydrothermal source and hydrographic
station.  For example, the simulation shows that prior to station~28 the upper
level flow was exclusively to the south and brought a plume from High Rise
(Figure~\ref{location-map}) to the NoMEF vicinity.  (Station 28 occurs from
MJD~51771, 20.4\,hr and MJD~51772, 6.5\,hr in the animation.)  As station~28
began (Figure~\ref{full_waterfall}), the High Rise plume was advected westward
by 10\,cm/s flow.  As the plume intensity decreased at NoMEF in response, the
flow decreased to near-zero.  The flow then increased to 7\,cm/s
south-southeast, ultimately delivering another pool of warm, particle-rich
fluid from High Rise to the NoMEF area near the end of the station.
Interpreted with respect to this flow history, it is plausible that the maximum
observed elevation of the upper plume boundary was attained (above a high $B$
High Rise source) when the cross flow was near zero.  

A similar progression of flow from south to west and back to south occurred
just before and during station~33 (MJD~51774, 2.5\,hr and MJD~51774, 17\,hr in
the animation).  Results from the puff model, with flow determined by the
record from FM-S250 and a single source at MEF, indicate that the SoMEF area
(Figure~\ref{abebathy}) should have been swept free of MEF plume about midway
through the station during the period of westward flow, but the observed plume
intensity increases during this period.  With the flow determined by the
observed current record from FM-S200 or FM-S150 (1992 or 2042\,m,
respectively), however, the area is swept free for a briefer period.  The
intensification of the observed plume during the period of flow to the
west-northwest could be explained if a source existed to the east-southeast of
station 33.  Hydrographic evidence of such a source was gathered in 1995
\cite{veirs+99} and confirmed during Flow Mow CTD surveillance between MEF and
Mothra.  This example illustrates the limitations of using the puff model with
a single source to predict or interpret plume distributions.  The modeled
distribution can be altered substantially by subtle changes in the input
velocity field, the addition of a source, model resolution, and level of
diffusivity.  For these reasons, I use the puff model primarily to study the
statistics of plume variability (\emph{c.f.}~Section~\ref{sec:modelupper}).  

One of the major features of the plume centered near 2000\,m is that it is
usually present and about 200\,m thick, but is intermittently replaced with
near-background cold, clear fluid.  
% good place to show NoSoMEF th--S diagrams!
Specifically, plume intensity decreases during stations 4 and 5, at the end of
14, the middle of 33, and the beginning and end of station 33.  A simple
explanation for this variability comes from the puff model.  Though it may not
simulate the plume distributions exactly, it does demonstrate that a point
source venting steadily into the typical flow above the ridge generally gives
rise to a heterogeneous distribution of plume and ambient fluid.

Another feature of the central plume is its vertical heterogeneity.
Particularly during station 5 and the beginning of stations 33 and 28, there
are distinct layers in the central plume.  Throughout station 5, similar upper
and lower plume layers are separated by background fluid.  During the first 2
hours of station 28, the upper portion of the plume has a
backscatter--$\Delta\theta$ ratio that is distinctly higher than the lower
portion of the plume.  A similar layering is evident in the first 5 hours of
station 33, but the lower plume is both warmer and richer in particles than the
upper plume.  These sorts of spatial variations are likely responsible for the
multiple trends observed in scatter plots of $\Delta_\rho\theta$ versus light
attenuation data from Endeavour plumes \cite[,~Figure~5]{baker+massoth87} and
from plumes within 1\,km of the MEF \cite[,~Figure~4]{thomson+92}.  While the
distinct trends have been attributed to vents with distinct source temperatures
and chemistries, in the case of station 28, the puff model indicates that the
responsible sources were probably all located within the High Rise field. 
% Do fluid chem props vary much at High Rise?

A final important feature of both time series in Figure~\ref{full_waterfall} is
that the anomalies of both $\theta$ and backscatter in the lower depths
($\sim$2100--2000\,m) are not clearly correlated with the $\theta$ and
backscatter anomalies in the overlying plumes.  Station~14 exhibits an intense,
uniform upper plume, but some of the smallest anomalies observed in the lower
layer.  Station~28 shows a similar anti-correlation, to some extent, with the
least intense upper plume near hour 14 accompanied by some of the largest
anomalies in the lower layer.  These anti-correlations might be expected given
the sheared flows between the 2 depth ranges and the likelihood that plumes in the 
2 layers originate from different points on the sea floor.  In contrast, 
stations 3--7, the end of 28, and the first half of station 33 exhibit trends in
anomalies that are similar in both the upper and lower layers.  These
positive correlations support the idea that anomalies in axial valley water are
enhanced, at least intermittently, by downward fluxes of heat, salt, and/or
particles from plumes passing overhead.

\begin{figure}
%\begin{sidewaysfigure} 
\begin{center}
\includegraphics*[width=.75\textwidth]{figs/upper/zbin_bks_hist_nomef_only.eps2}

%\includegraphics*[width=.45\textwidth]{figs/upper/all_zbin_bks_hist_nomef.eps2}
\includegraphics*[width=.75\textwidth]{figs/upper/zbin_bks_hist_somef_only.eps2}
%\hfill
%\includegraphics*[width=.45\textwidth]{figs/upper/all_zbin_bks_hist_somef.eps2}
\caption[Depth-binned histogram of backscatter $\sim$500\,m north and south of
ME]{Histograms of all backscatter data from the NoMEF (upper) and SoMEF (lower)
areas (Figure~\ref{abebathy}), except station 7, binned in depth intervals of
25\,m.  Mean values of distribution in each bin are listed at right. 
%
%The 2 panels on the left incorporate CTD data from the NoMEF and SoMEF
%stations ($\sim$500~m north and south of the MEF boundaries).  The right hand
%panels %incorporate additional data from closer CTD stations: those that
%surveyed the north and south control surfaces.
} \label{bks_hist} \end{center}
%\end{sidewaysfigure}
\end{figure}

The differences in particle concentration profiles from the NoMEF and SoMEF
areas can be further illustrated with depth-binned histograms of the
backscatter data
%, first from the more distant stations alone, and then from both distant and
%near-field stations 
(Figure~\ref{bks_hist}).  In the bottom 100\,m the fluid is more uniformly
clear south of the MEF.  At both NoMEF and SoMEF, backscatter values in the
overlying plume are higher than near the bottom.  The peak values tend to occur
higher in the water column north of the MEF, a pattern that is evident in
Figure~\ref{ns_surfs}, as well.  The mean values between 1900--2100\,m,
however, are similar between NoMEF and SoMEF, suggesting that the difference in
backscatter values near the bottom is not due to a greater downward flux of
settling hydrothermal particles at NoMEF than at SoMEF.  An alternative
possibility is that hydrothermal particles (mineral or biological) in diffuse
or focussed plumes from MEF sources are predominantly transported northward by
the mean flow in the bottom 50--100\,m.  This interpretation is consistent with
the average pattern of near-bottom $\Delta\theta$ being higher at NoMEF than at
SoMEF.

%\subsection{North, south, and east control surfaces: 1800--2200\,m}
\subsection{North and south control surfaces: 1800--2200\,m}
\label{sec:ns_surfs}

Figure~\ref{ns_surfs} showcases the south and north control surfaces during 3
separate navigated CTD stations (10--13), each a VOT that traversed the surface
1--5~times from 1800--2200\,m (\emph{c.f.}~Section~\ref{sec:mef_dth_lower}).
Note that these surfaces extend from above the ridge, higher than typical rise
heights of MEF high $B$ sources, to within the axial valley, $\lesssim$5\,m
above the sea floor.  This depth range spans the observed vertical shear
between the axial valley flow and the currents above the ridge crests.

Closer to the MEF, the observed $\Delta\theta$ and backscatter fields on the
north and south control surfaces in Figure~\ref{ns_surfs} reveal a similar
pattern of a sharp upper plume boundary, a 100-200\,m thick plume, and an
underlying layer of cooler, clearer fluid (Figure~\ref{ns_surfs}).  But the
spatial horizontal and vertical heterogeneity is noticeably increased compared
to levels observed on VOTs through the more distant SoMEF and NoMEF areas.  On
these surfaces, laterally offset just 100--200\,m from the MEF high $B$
sources, background fluid is often interspersed with intense plumes near the
equilibration depths.  In such cases, observed across-valley gradients ($\dif
\Delta\theta/\dif x\simeq$0.1/50$\sim2\times10^{-3}$\,$^\circ$C/m) begin to
rival vertical gradients ($\dif \Delta\theta/\dif
z\simeq$0.1/10$\sim1\times10^{-2}$\,$^\circ$C/m).  The upper plume boundary is
again quite consistent, $\sim$1870$\pm$20\,m, but is slightly shallower on
average than observed during the NoMEF and SoMEF time series.  It is poorly
defined on some surfaces (stations 10\#1, 12\#2, and 12\#3) and exceptionally
sharp during station~13.  

\begin{figure}
\begin{center} 
\includegraphics*[width=.70\textwidth]{figs/upper/allctd_NSwalls_fullz.eps2}

\includegraphics*[width=.70\textwidth]{figs/upper/allctd_NSwalls_fullz_bks.eps2}
\caption[$\Delta\theta$ and backscatter distribution north and south of MEF
(1800--2200\,m)]{$\Delta\theta$ (upper panels) and backscatter (lower panels)
distribution on north and south MEF control surfaces: 1800--2200\,~m.  Left 2
columns depict surveys of south surface; right column depicts surveys of north
surface.  Each survey is labeled with: its station number; its order in the
sequence of surfaces surveyed during that station (\emph{e.g.}~12\#3 is the 3rd
surface observed during station 12); and the mean value of $\Delta\theta$ or
backscatter (noted parenthetically).  White areas have no data (or
$\Delta\theta<0$); black lines are CTD track.  Central color scales give
values.} \label{ns_surfs} \end{center} \end{figure}

The main plume between $\sim$1850--2150\,m is remarkably less homogeneous than
in the time series from the more distant NoMEF and SoMEF stations.  On all
surfaces except station~13, not only does the main plume contain finer scale
heterogeneities (order 10\,m thick and 50\,m wide), but also it is more layered
and laterally patchy at scales of 50--100\,m.  During station~12, background
fluid is more prevalent on the west edge of the southern surface, while on the
northern surface (station~10), background fluid is more common on the eastern
edge.  In general, the plume distribution on the northern surfaces is less
continuous, both laterally and vertically.  Consequently, the mean
$\Delta\theta$ on each of the northern surfaces is lower than the mean on any
of the southern surfaces.  This is consistent with the pattern predicted by the
puff model.  Station~10 
%MJD~51762.85--51763.16 
occurred during a transition from southward to westward flow in which the model
suggests plumes from High Rise were advected through the western side of the
northern MEF control surface.  A few hours later, the upper level flow was
unidirectional and to the south-southeast during station~12, bringing the MEF
plumes through the south surface and resulting in elevated mean values.  
%(MJD~51763.32--51763.67).  
The next day, south-southwest flow before and during station~13 caused the
maximum observed mean values in Figure~\ref{ns_surfs}.
% (MJD~51764.26--51764.38).
%
% The uniform southward flow during each of these stations might motivate
% another set of estimates based on the mean flow and mean S-N $\Delta\theta$
% values... Need to adjust for unsampled area, however, and note that westward
% flow near end of station 10 may bias the H_h estimate because in consistent
% southward flow more of the HR plume would likely warm the N surface...
%
% Also, is it possible that the sward flow during station 12/13 added heat to
% the lower layer, despite Nward near-bottom flow ?  Could this help explain so
% many warmer S surfaces in lower H_h calc, despite mean warming to N?  Why did
% the lower plume edge become less well defined during station 12?  There's
% hardly any lower layer...

With increasing proximity to the MEF, the lower layer is thinner and warmer
than at the more distant stations.  While the backscatter data generally
suggest the overlying plume is still separated vertically from the sea floor by
50--100\,m, the $\Delta\theta$ data from station~12 indicate the separation on
the south surfaces is only 10--40\,m.  In some places, near-bottom anomalies
make the lower layer ill-defined (\emph{c.f.}~section~\ref{sec:mef_dth_lower}).
During station~13 a cooler, clearer layer is evident and separation is closer
to 50--100\,m.  On the north surfaces a lower layer (2050--2200\,m) is warm and
particle rich relative to the overlying background fluid on the east edge, but
essentially continuous with the overlying plume on the west edge.
Additionally, the north surfaces are distinct from most of the south surfaces
in that the $\Delta\theta$ increases into the bottom along a greater portion of
their bottom edge. 
% This could be offered up as additional subtle evidence of diffuse plumes
% being advected N near bottom...

% Add Stn 20 VOT time series (or 3d curtain view?), or maybe just use th-s plot?
% /setting/depth_bin_dth_hist_stn20.10m.gt2100.eps2 ?
% 
% if you add, then change section title...


\subsection{Consecutive surveys of the MEF perimeter: 1800--2070\,m}

At the end of the Flow Mow field program, the perimeter of the upper control
volume was slowly circumnavigated counterclockwise $\sim$5 times during a
$\sim$24\,hr VOT.  The VOT focused on the depths from 1850--2070\,m in an
effort to intercept equilibrated or equilibrating MEF plumes.  During the same
period, ABE surveyed the top surface of the lower control volume near 2110\,m,
acquiring 2~estimates of the vertical heat flux \cite{stahr+03}.  
% What was mean H_v for those 2 tops, only??

In this section I present the data from the circumnavigations.  In the
following section the observations are used to assess the net horizontal heat
flux through the upper control volume, which is then compared to the vertical
heat flux input from below.

\begin{sidewaysfigure} \begin{center}
\includegraphics*[width=.75\textwidth]{figs/upper/Dth35-39_annot.eps}
\caption[$\Delta\theta$ on 5 circuits of MEF perimeter: 1850--2070\,m]{
$\Delta\theta$ observed during 5 counterclockwise circuits of MEF perimeter
starting on the west surface during station~35 and oscillating through
1850--2070\,m.  Data are projected orthogonally onto control surfaces (width
$L$ is 300 or 700\,m, height $Z$ is 225\,m) and averaged in grid cells that are
50\,m wide and 5\,m tall.  The south, west, north, and east side surfaces are
displayed as if a continuous ribbon were broken at the southeast corner of the
MEF and folded open (see schematic at bottom).  The title above each surface
labels: the CTD station number (35--39); the number of the surface surveyed
sequentially during each station; the cardinal direction of the surface (S, W,
N, or E); and the mean $\Delta\theta$ (noted parenthetically).  Each column
constitutes a time series of the same surface, with temporal separation of
$\sim$4--4.5\,hr.  Color scale is the same as in Figure~\ref{full_waterfall}.
Black line is the CTD track.  } \label{ribbon} \end{center}
\end{sidewaysfigure}

% Where's the backscatter version of this plot?

Figure~\ref{ribbon} displays the 5~circumnavigations, illustrating the spatial
and temporal variability in $\Delta\theta$ on the 220\,m tall side surfaces
(1850--2070\,m) of the upper control volume (Figure~\ref{fmcv}).  Overall, the
temporal variability is high and suggestive of semi-diurnal oscillations, with
sequential surveys of particular side surfaces often showing dramatic changes
in plume distributions despite being separated by only 4--4.5\,hr.  The north
surfaces, for example, alternate between near background fluid at all depths
and laterally-continuous plumes with peak $\Delta\theta$ values that are
comparable to the highest values on the west, south, or east surfaces.  A
single south surface is also nearly completely plume-free, while the previous
and following surveys of the same surface ($\sim$4.5\,hr before and after) are
dominated by plumes.  The east and west surfaces are comparatively consistent
between surveys.  Both tend to have higher mean $\Delta\theta$ than the north
and south surfaces, and the west surfaces have the highest mean $\Delta\theta$,
overall.
 
The upper boundary of the plume is deeper and more variable than in
Figures~\ref{full_waterfall}~and~\ref{ns_surfs}, with a mean depth of
$\sim1950\pm$40\,m.  On some surfaces the plume is homogeneous
(surfaces~38\#3~and~39\#1), while on many it is horizontally and vertically
patchy (\emph{e.g.}~surfaces~37\#3--4).  On surface~36\#1, the main plume is
almost as narrow as it is thick, $\sim$100\,m, and surfaces~38\#4~and~39\#1
captured an unusually thin plume near 1900\,m that is clearly separated from
the underlying plume. 

On each surface that is free of major plumes the bottom edge is relatively
warm.  At a depth of 2070\,m the bottom edge is near the depth of the west
ridge crest adjacent to the MEF (Figure~\ref{parallel_bathy}), but is not
within the axial valley.  Thus, higher $\Delta\theta$ values along the bottom
edges of the upper control volume surfaces suggest that fluid near the top of
the axial valley is heated in part by a source other than overlying plumes.
Currents measured at 2092\,m by FM-S100 indicate that fluid near the bottom
edge of the upper control volume is also shielded to some degree from advection
off-axis by the overlying currents (Figure~\ref{5level_pvd}), and may therefore
retain plumes even when higher plumes are swept away.
  
The upper edge of the control volume at 1850\,m was consistently well above the
loftiest plume.  This supports the assumption that there was no vertical heat
flux through the top of the upper control volume.  In contrast, only
occasionally is a lower plume boundary evident on the surfaces above the lower
edge (2070\,m).  It is therefore possible that some of the horizontal heat flux
was not assessed because plume fluid escaped through the $\sim$30--40\,m gap
between the bottom of the upper control surfaces and the top of the lower
control volume where ABE measured $H_v$.  This possibility is discussed further
below.


\section{Calculated horizontal heat flux}

Calculation of horizontal heat flux through the upper control volume requires
the $\Delta\theta$ observed during the circumnavigation stations and a source
of current measurements.  The southern current meters at 1942, 1992, and
2042\,m depth (FM-S150, FM-S200, and FM-S250, Table~\ref{cm_table}) provide the
best available measurements of the flow through the upper control volume during
the Flow Mow field program.  A PVD for any of the 3 upper meters indicates that
during the Flow Mow cruise prior to the circumnavigation stations
(MJD~51760--51774), the mean flow was primarily to the southwest
(Figure~\ref{5level_pvd}).  Examination of the puff model based on the record
from any of the 3 meters shows that during the same period before the
circumnavigation stations, the mean flow moved plumes to the southwest overall,
but oscillatory flow caused the plume to undulate between the west and
southwest directions.  

During the period of the 5 CTD circuits around the MEF
(MJD~51775.60--51776.52), the flow exhibited relatively unusual variations
compared to the majority of the current record.  Near the end of the Flow Mow
cruise when the circumnavigating CTD stations were undertaken, and for a few
days after the cruise ended, the mean flow weakened substantially and the
oscillatory component strengthened and became more rotary.
Figure~\ref{curr35-39} shows the hourly current sequence that occurred at each
of the 3 upper meters during the circumnavigation stations (35--39).  Flow at
the 2 deeper meters (1992 and 2042\,m) was very similar; it was initially
$\sim$8\,cm/s to the southeast, then weakened to $\sim$1\,cm/s, shifted
clockwise to the southwest, and increased to $\sim$4\,cm/s.  Finally, it
decreased through zero, rose to $\sim$3\,cm/s northeastward, and ended by
decreasing in magnitude as it rotated clockwise.  At FM-S250  (1942\,m) the
sequence of flow direction mimicked the deeper 2 meters, but the magnitudes
remained above 5\,cm/s.

\begin{figure} \begin{center}
\includegraphics*[width=.75\textwidth]{figs/upper/curr35-39.eps2} \caption[Mean
currents during 5 circuits of MEF: 1850--2070\,m]{ Time series of
quarter-hourly mean currents observed during 5 circuits of the upper control
volume at depths of 1942\,m (top), 1992\,m (middle), 2042\,m (bottom) on the
southern mooring (FM-S, Table~\ref{cm_table}).  Vertical dashed lines show the
time at which the survey of each surface began during stations 35--39 (see
Figure~\ref{ribbon}).  Each station label is just to right of time at which
first surface survey began for each station. } \label{curr35-39} \end{center}
\end{figure}

Faced with this unusual period of current direction changes and potential
pooling events, and indications from the puff model that plumes from High Rise
may have been entering the MEF during the circumnavigations, I endeavor in this
section to characterize the net horizontal heat flux through the upper control
surface through 2 distinct methods.  First, I estimate the heat flux through
each individual surface by averaging the $\Delta\theta$ and $u$ data.  These
fluxes are then averaged and summed to estimate the mean net flux.  Second, I
compute 16 estimates of ``quasi-synoptic'' net flux by summing the heat flux
through all available sets of 4 sequential surfaces.


\subsection{Mean net horizontal flux}

I begin the process of calculating the mean net horizontal flux
$\overline{H_h}$ by estimating the mean heat flux $H_h$ for each of the 19
surfaces in Figure~\ref{ribbon}.  
%This is accomplished by first weighting the mean $\Delta\theta$ on each
%surface by the ratio of area sampled to total surface area.  
This is accomplished by first computing the mean $\Delta\theta$ on each
surface, $\overline{\Delta\theta}$.  After spatial averaging of the raw data in
5$\times$50\,m cells, 65--87\% of the cells contain data.
%If the data gaps were assumed to have $\Delta\theta=$0, one could adjust
%$\overline{\Delta\theta}$ according to the ratio of area sampled to total
%surface area.  However, 
These data gaps are distributed about equally along the warm lower edge and
cool upper edge of each surface (Figure~\ref{ribbon}) and are most likely to
have $\Delta\theta$ similar to adjacent cells.  Consequently, the data gaps are
assumed to have $\Delta\theta=\overline{\Delta\theta}$ and
$\overline{\Delta\theta}$ for each surface is therefore left unadjusted.  
% A similar effect could be accomplished by increasing the cell size slightly.  
Next, the normal component (outward positive) of the mean $\mathbf{u}$ observed
during the traverse of each surface is assessed using the current record from
FM-S200 (1992\,m).  While FM-S250 at a depth of 1942\,m is more central to the
upper control surface depth range (1850--2070\,m), I elect to use FM-S200
because it recorded intermediate speeds --- slower than the flow 50\,m above at
FM-S250, but faster than flow 50\,m below at FM-S150 -- and therefore will
generate more reasonable estimates of the horizontal heat flux through the
entire surface.  
% This suggests an an alternative set of heat fluxes, or the associated error
% should be discussed somewhere!  It also indicates that a more complicated
% method of multiple depth ranges is being passed up...
Since some surfaces were traversed in $\lesssim$1\,hr, the FM-S150 was
interpolated from 1\,hr samples to 0.25\,hr samples in order to obtain an
accurate estimate of mean $\mathbf{u}$ for each surface.  Finally, the
orthogonal component of mean flow $\overline{u}$ was computed and multiplied by
$\overline{\Delta\theta}$, $\rho
c_p=4.2$\,MJ$\cdot$m$^{-3}\cdot^\circ$C$^{-1}$, and surface area $A$ of each
respective surface, yielding an estimate of $H_h$ for each surface.  

In a steady cross flow typical of the equilibration depths ($u\sim$5\,cm/s),
the heat flux through a single MEF control surface with
$\overline{\Delta\theta}\simeq0.04^\circ$C (Figure~\ref{ribbon}) should be
approximately
\begin{equation}
H_h=\rho c_p \overline{u} \overline{\Delta\theta} A \simeq
%(4.2\times10^6)(0.05)(0.04)(220\times L), 
1.85 L,
\label{rough_upper}
\end{equation}
where $A$ is the product of $Z$, the vertical extent of the surface (220\,m)
and $L$, the length of the surface (either 300 or 700\,m).  Using $L$ of 300
and 700\,m, $H_h$ is 550 and 1300\,MW, respectively.  An increase of
0.01~units in $\overline{u}$ or $\overline{\Delta\theta}$ increases $H_h$ by
$\sim$20\%.  

The observed values of $H_h$ have absolute magnitudes of 0--1850\,MW and are
presented in Figure~\ref{allseries35-39} for each of the 19 surfaces, along
with the associated sequences of $\overline{u}$, $\overline{\Delta\theta}$, and
data coverage as a percent of total surface area, $A$.  Positive fluxes are
oriented outward for any given surface.  The peak magnitudes of $H_h$, about
$-1000$ and $+2100$\,MW, were observed on the west and east surfaces,
respectively (sides 1~and~3 of station~35 in Figure~\ref{ribbon}), indicating
eastward advection of the $\Delta\theta$ distribution.  
% confirmed with the puff model?
The $\overline{\Delta\theta}$ on these 2 surfaces was not exceptional, but the
mean flow to the south--southeast was consistent and relatively strong
($\sim$5--7\,cm/s).  The magnitude of the peak $H_h$, $\sim$2100\,MW, is
$\sim$1/3 the estimates by \citeN{thomson+92} of ``instantaneous'' flux through
a single cross section of the MEF plume core: 6300~and~6800\,MW.

\begin{figure} \begin{center}
\includegraphics*[width=.75\textwidth]{figs/upper/Allseries35-39.eps2}
\caption[Flow, $\Delta\theta$, data coverage, and heat flux through individual
surfaces]{Time series of observations made on individual side surfaces during
circumnavigation of the upper control volume (Stations~35--39).  Panels (top to
bottom) portray: mean orthogonal $u$; mean $\Delta\theta$; data coverage as
percent of total surface area; and heat flux (positive outward).  Lines connect
sequential observations made on the same side surface and are labeled only in
upper panel: circles=south; x=west; +=north; triangles=east.  Horizontal axis
labels indicate  the station number and surface number during that station,
separated by a decimal point (station.surface).  } \label{allseries35-39}
\end{center} \end{figure}

The observed ranges of $\overline{u}$, $\overline{\Delta\theta}$, and percent
data coverage in Figure~\ref{allseries35-39} are -3.5--7.5\,cm/s,
0.001--0.058$^\circ$C, and 67--89\%, respectively.  It may seem surprising that
the maximum $\overline{\Delta\theta}$ observed during circumnavigations of the
upper control volume is 0.058$^\circ$C when $\overline{\Delta\theta}$ is higher
on many north and south surfaces of the lower control volume
(\emph{c.f.}~Section~\ref{sec:mef_dth_lower}).  This occurs because the
$\Delta\theta$ sample population on the surfaces of the upper control volume
contains many near-zero values that reduce $\overline{\Delta\theta}$, despite
the presence of many samples with $\Delta\theta$ much higher observed on the
lower control surfaces.  As with other data collected near the equilibration
depths of MEF plumes, the variance of $\Delta\theta$ sampled during the
circumnavigation stations is elevated compared to data from the surfaces of the
lower control volume (Figure~\ref{depth-hist}), reflecting the multidirectinal
versus rectilinear nature of currents above versus within the axial valley.

%The standard deviations of the sequences, grouped by side surface, indicates
%that a primary source of variance is... (Table~\ref{}).  (Use these variances
%to estimate error in mean and/or compare to statistics of modeled flux?)

To estimate the net mean horizontal flux $\overline{H_h}$, the values of $H_h$
for the individual surfaces are grouped by cardinal direction (the 4 groups of
connected points in the bottom panel of Figure~\ref{allseries35-39}).  Each
group of values is then averaged to yield the group mean flux $\overline{H_g}$
through each side of the control volume.  The 4 resultant magnitudes are
tabulated in Table~\ref{netupperflux} and summed to yield
$\overline{H_h}=$296\,MW.
%\pm1713$\,MW.

\begin{table}

\caption[Net flux through the MEF vertical control surfaces near 1942\,m] {The
average and standard deviation of the sequence of horizontal heat flux observed
on each surface of the MEF control volume.  The group mean fluxes
$\overline{H_g}$ and ${\overline{H_h}}_j$ are calculated slightly differently,
as explained in the text (see Equation~\ref{hhmeansum}).  The sequences are
portrayed in the bottom panel of Figure~\ref{allseries35-39}.  The right hand
column lists the total, or net heat flux and the sum of the standard
deviations.}

\label{netupperflux}

\begin{center}
\begin{tabular}{lrrrrr}
Side surface         & South  & West   & North   & East      & Total, $\overline{H_h}$\\
\hline
$\overline{H_g}$     &    109 &$-$334  &     32  &   489  &  $+$296\,MW\\
Standard deviation   &    187 &   417  &    159  &   950  &  
%$\pm$1713\,MW 
\\
${\overline{H_h}}_j$ &     90 &$-$358  &  $-$24  &   339  &  $+$47\,MW\\
\end{tabular}
\end{center}
\end{table}

%In analogy with the approach taken to estimate the mean net horizontal flux in
%the lower control volume (Chapter~\ref{ch:lower}), a
An alternative mean net horizontal flux in the upper control volume can be
derived by finding the mean values of $\overline{u}$ and
$\overline{\Delta\theta}$ for each group first, and then multiplying according
to \begin{equation} {\overline{H_h}}_j = \rho c_p {\overline{u}}_j
{\overline{\Delta\theta}}_j A, \label{hhmeansum} \end{equation} to estimate the
mean flux through each of the $j$ side surfaces.  The resulting values are
listed in the last row of Table~\ref{netupperflux} and sum to give a different
estimate of $\overline{H_h}=$47\,MW.
% Real analogy with ch:lower is taking mean u for entire circumnav (WAHt is
% it?!?!) and using it with mean dTh for each group  to get 4 H_h, which summed
% yields mean...

% Could also take historic approach of using long-term mean (all fm cruise?) (all south 
% mooring record?) and error estimate from the currents only...  Could combine that 
% uncertainty with the observed variance of dth on the surfaces... 

% Later, use puff model to put an error estimate on this?  Or could use rms(u) technique...

The discrepancy between these alternative estimates of $\overline{H_h}$ implies
that the instantaneous net heat flux through the upper control volume varies on
short time scales and yields different mean values depending on the averaging
period.  This suggests that an accurate value of $H_h$ depends on longer
averaging periods and/or sampling more synoptically and frequently.  Since the
overall duration of the hydrographic observations (circumnavigation stations
35--39) is limited to only a single day, my approach is to treat the available
observations as nearly synoptic and use them to approximate an instantaneous
heat flux series.


\subsection{Quasi-synoptic horizontal flux}

Full resolution of the variations in net heat flux through the MEF perimeter
over the equilibrium depths would require synoptic monitoring of $\Delta\theta$
and $\mathbf{u}$ at all points on all side surfaces.  During the
circumnavigations of the upper control volume, the current field was resolved
much more coarsely (at a single point $\sim$1\,km to the south) and the
$\Delta\theta$ distribution was assessed on each surface only once every
4--4.5\,hr.  Given the variability of the $\Delta\theta$ field and the dominant
current oscillation period of 12\,hr, the time required to circumnavigate the
upper control volume is still undesirably slow.  Nevertheless, it is possible
to assume each set of 4 sequential surfaces is an instantaneous and synoptic
observation and to calculate a ``quasi-synoptic'' flux, $H_{qs}$, by summing
the fluxes through each side.  Using all 19~surfaces, a series of 16 (=19+1-4)
estimates of $H_{qs}$ is obtained that describes approximately the temporal
evolution of the net horizontal heat flux through the upper control volume.

The 16 estimates of $H_{qs}$ are generated by first calculating $H_h$ for each
of the 19 surfaces, as described in the previous section, and then summing over
each unique set of 4 sequential side surfaces, according to \begin{equation}
{H_{qs}}_j = \sum_{i=j}^{j+4} {H_h}_i \label{hhsum} \end{equation} in which $j$
proceeds from 1, when the first 4 surfaces are summed, to 16, when the final 4
surfaces are summed.  The resulting series of estimates is displayed in
Figure~\ref{quasiplot}, along with the mean $\overline{H_{qs}}$.

\begin{figure} \begin{center}
\includegraphics*[width=.75\textwidth]{figs/upper/Hseries35-39.eps2}
\caption[``Quasi-synoptic'' and mean net heat flux through the upper control
volume] {Series of 16 estimates of the ``quasi-synoptic'' net horizontal heat
flux $H_{qs}$ through the upper MEF control volume.  The mean magnitude
$\overline{H_{qs}}$, is overlain for reference.  Each estimate was derived by
summing a set of 4 sequential surfaces from the 19 total observed surfaces (see
Equation~\ref{hhsum}).  The mean, standard error of the mean, and standard
deviation of the quasi-synoptic estimates are 378, 182, and $\pm$730\,MW,
respectively.} \label{quasiplot} \end{center} \end{figure}

The first few estimates have the largest magnitudes (1400--2000\,MW) because of
the relatively high $\overline{u}$ and $\overline{\Delta\theta}$ observed on
the first 3~surfaces during the first circumnavigation
(Figure~\ref{allseries35-39}).  Subsequent estimates are between $\pm$600\,MW.
Averaging the 16~estimates yields the mean quasi-synoptic net flux,
$\overline{H_{qs}}=$378$\pm$730\,MW.  The standard error of the mean is
$730/\sqrt{16}\simeq182$\,MW, indicating an uncertainty of $182/378\simeq48$\%.
The puff model analysis in the next section illustrates that even if the side
surfaces were instrumented to continuously monitor $\Delta\theta$ and $u$ at
all points, the standard deviation of the instantaneous net $H_h$ would be
surprisingly high, requiring longer observational periods to achieve more
precise estimates of the mean horizontal flux. 


\section{Modeled horizontal heat flux}
\label{sec:modelupper}

The puff model simulations for the equilibration depths were conducted with the
same puff model used for the lower control volume
(\emph{c.f.}~Section~\ref{sec:puffmodel}), but the input flux was set to
600\,MW and the flow was determined by \emph{observed} current records from
above the ridge crest.  The MEF perimeter, horizontal diffusivity, and initial
puff dimensions were held constant.  In some cases, grid resolution and time
step interval were altered to optimize computational efficiency.

In this section I present puff model runs based on the longest available
current meter records, with attention to the statistics of the net heat flux
time series that is generated by the observed flow.  I also use the puff model
to consider whether plumes from non-MEF sources may have influenced the upper
control volume during the circumnavigation stations.

\subsection{Modeled variance in flow above ridge crests}

The puff model with upper level flow is interesting to watch because of the
complexity of distinct plume distributions that are generated during different
combinations of oscillatory and mean flow.  The
\href{http://econscience.org/scott/pubs/thesis/cm200.full.fli}{simulation
based on FM-S200} at 1992\,m, for example, has long periods where the plume
streams fairly steadily and unidirectionally, and occasional stretches where
the plume stalls and oscillates directly over the MEF
(Figure~\ref{upperpuffframes}).  

\begin{figure}
\begin{center} 
\includegraphics*[width=.75\textwidth]{figs/puff/upperstream.eps}

\includegraphics*[width=.75\textwidth]{figs/puff/upperpool.eps}
\caption[Examples of modeled plume distribution: streaming and
pooling]{Individual ``snapshots'' or frames from the puff model animations
illustrating streaming (upper) and pooling (lower) plumes.  Flow is determined
by the full record from FM-S200, the current meter at 1992\,m on the southern
mooring.  See Figure~\ref{puff-model} for additional details.}
\label{upperpuffframes} \end{center} \end{figure}

While the typical situation during long current meter records is a plume that
is advected on average to the southwest as it whips between south and west
directions, the plume distribution exhibits a wide range of alternative
evolutions.  (Links to animations:
\href{http://econscience.org/scott/pubs/thesis/cm200.full.fli}{FM-S200 at
1992\,m};
\href{http://econscience.org/scott/pubs/thesis/cm250_long.fli}{FM-S250 at
1942\,m}; \href{http://econscience.org/scott//pubs/thesis/mz300.fli}{MZ300 at
1900\,m}.)  Advection of these diverse and heterogeneous $\Delta\theta$
distributions through the control surfaces leads to a high standard deviation
of the instantaneous net heat flux (displayed in uppermost panel of animation
frames).  For the full record of FM-S200, the standard deviation is
$\pm$800\,MW, or $\sim$130\% of the 600\,MW input flux.  The peak magnitude of
the modeled instantaneous heat flux is much greater, ranging from
$-$3800~to~$+$8700\,MW.
%  The standard deviation for the full record of the Mixing Zephyrs (1995)
%  meter at 1900\,m is $\pm$\%.  

The standard deviation of the time series of modeled net heat flux is lowest
during consistent streaming periods, moderate during pooling events, and
highest during intermediate combinations of oscillation and mean flow.  In the
intermediate type of flow, the peak instantaneous flux occurs when warm pooled
water is advected through only a \emph{single} surface, leading to a large net
flux because the outward downstream flux is not offset by an inward upstream
flux.  Additionally, the velocity associated with flushing of the pool out one
side of the volume is generally higher than the typical flow speed during the
pooling event.  In contrast, the standard deviation is lower when oscillations
dominate mean flow and generate a plume larger than the control volume.  Within
such a plume $\Delta\theta$ distributions are similar on both upstream and
downstream surfaces, so inward and outward fluxes tend to balance, resulting in
lower variance in the net flux.  The variance in net flux is also relatively
low in streaming flow because $\Delta\theta$ varies less on the downstream
surface than when oscillations are strong enough to cause pooling.

During the Flow Mow hydrographic survey period (MJD 51761.0--51776.6; see
Appendix~\ref{ap:dates}) the animation of the model with flow determined by
FM-S250 shows a plume advected along-axis to the south-southwest on average,
but intermittently transported up to $\sim$1\,km in directions between
west-northwest and south-southeast.  A reasonable initial estimate of the
standard deviation of the observed net flux through the upper control volume is
the standard deviation of the modeled net flux based on the entire FM-S200
record: $\pm800$\,MW.  
%Then get to variance during ribbon survey.  Implement iterative scheme as with
%observations.  Compare modeled and observed mean and variance, relative to
%input flux.

One way to refine the expected standard deviation of the observed
$\overline{H_h}$ is to analyze the mean and standard deviation of the modeled
instantaneous net flux.  From the standpoint of designing methods to
measure $\overline{H_h}$, a useful approach is to examine how the statistics of
the time series changes with the length of the observational period (often the
key constraint on a field program).  I do this by breaking the full simulation
period ($\sim$1800\,hr) into a range of shorter periods (1, 5, 10, 15, 25, 50,
100, 250, 500, and 1000\,hr) and computing the standard deviation and mean of
each sub-series.  The average value of the standard deviation, taken over all
available sub-series of a given period, is presented as a function of the
observation period in Figure~\ref{std_dur}, using data from the model forced
with the record from FM-S200 (1942\,m).   

\begin{figure} \begin{center}
\includegraphics*[width=.75\textwidth]{figs/puff/singlesurf_std.eps2}
\caption[Standard deviation of modeled flux as a function of observation
period]{Standard deviation of modeled flux as a function of duration of
observation period.  Curves are shown for the net instantaneous flux, as well
as the instantaneous flux through individual surfaces (south, west, north, and
east).  Data are from puff model with currents from FM-S200 and input flux set
to 600\,MW.} \label{std_dur} \end{center} \end{figure}

This main insight from this analysis is that the standard deviation of
instantaneous net heat flux is expected to decrease rapidly with the duration
of synoptic observations.  While the average standard deviation is $\sim$100\%
of the input flux for a 5\,hr observational period, for a slightly longer
observational period like the $\sim$25\,hr circumnavigation survey, the average
standard deviation of the modeled net instantaneous flux drops to
$\sim$100\,MW, or about 17\% of the 600\,MW input flux.  Assuming that the
input flux is the observed $H_v\simeq$550\,MW \cite{stahr+03}, and that the
estimates from the quasi-synoptic circumnavigation survey represent
instantaneous flux estimates, the average standard deviation of the observed
$\overline{H_h}$ is expected to be $\sim\pm$92\,MW.  Note also that the curves
for individual surfaces differ dramatically, implying that observations of
$H_h$ on a single surface will yield a mean value \emph{and} standard deviation
that depend on which surface is surveyed. 


\subsection{Plumes from other vent fields}


Another result from the puff model is a synoptic simulation of the changing
$\Delta\theta$ distribution on each surface of the upper control volume.  Based
on the FM-S250 current record, Figure~\ref{ribbon_model} shows a time series of
the modeled $\Delta\theta$ distribution on each control surface, spread flat as
in the presentation of the observed $\Delta\theta$ distributions
(Figure~\ref{ribbon}).  

\begin{figure} \begin{center}
\includegraphics*[width=.70\textwidth]{figs/upper/cm250.ribbon.dT_mjd.full.eps}

\includegraphics*[width=.70\textwidth]{figs/upper/cm250.ribbon.dT_mjd.35-39.eps}
\caption[Modeled $\Delta\theta$ on MEF perimeter]{Time series of $\Delta\theta$
on the 4 MEF control surfaces, based on the current record from FM-S200.
Vertical axis is time in days with model time steps of 1/2\,hr; horizontal axis
is distance along successive control surfaces.  Upper panel shows the full
current meter record (Flow Mow cruise period is MJD~51760--51777); lower panel
shows only the period of the circumnavigation stations
(MJD~51775.60--51776.52).  Horizontal white lines mark the beginnings of CTD
stations~35--39, while vertical white lines represent boundaries between the 4
control surfaces (folded flat as in Figure~\ref{ribbon} and labeled with a
letter denoting cardinal direction).} \label{ribbon_model} \end{center}
\end{figure}

The full simulation record (Figure~\ref{ribbon_model}, upper panel) shows that
during the Flow Mow cruise period (MJD 51761.0--51776.6) the horizontal heat
flux associated with the MEF sources was most commonly through the west
surface, as it was for $\sim$10~days prior to the cruise.  Using current data
collected after the cruise, however, the model indicates the maximum flux was
through the east or south surfaces.  Significant fluxes through the north
surface occurred only during 3 short periods (near MJD~51772, 51783, and
51794).  In contrast, the available observations of the north surface show
non-zero heat fluxes at other times, too.  The model simulates neither the
2~north surfaces with non-zero fluxes observed during the circumnavigation
period (Compare Figures~\ref{ribbon} and lower panel of
Figure~\ref{ribbon_model}) nor the non-zero fluxes through the upper portion of
the 3~north surfaces observed during station~10 (Figure~\ref{ns_surfs};
MJD~51762.85--51763.16).  The simulation does show heat flux through the north
surface during NoMEF station~28 (MJD~51771.85--51772.27), when positive
$\Delta\theta$ was recorded near the equilibration depths
(Figure~\ref{full_waterfall}).

During the circumnavigations of the upper control volume, puff model
simulations based on flow at FM-S200 and FM-S250 both suggest that plumes from
High Rise, and possibly other upstream hydrothermal sources, were likely
advected through the Flow Mow control volume along with MEF plumes.  Such
juxtaposition of plumes is probably common at all but the northernmost vent
field on the Endeavour segment, given that the southwestward mean flow above
the crests is approximately aligned with the strike of the ridge and fields are
distributed along the axis.  

The likelihood of plume juxtapositioning underscores the importance of
monitoring all 4 sides of the control volume when attempting to measure the net
horizontal flux accurately over the Endeavour segment.  An upstream source will
generate hydrographic variability that will increase the variance in
$\overline{H_h}$ when the plumes from the upstream source traverse the control
surfaces.  Thus, the estimated standard deviations presented in the previous
section are most likely minima because the puff model does not include upstream
sources.  

\section{Discussion}

\subsection{Sources of hydrographic variability}

Considered together, the Flow Mow observations of hydrothermal hydrography
above the ridge and near the MEF describe heterogeneity at scales as short as
10--50\,m and 1\,hr and high variability at the scale of the MEF
(100--1000\,m).  These non-uniform distributions of the equilibrating plumes
imply that mesoscale vortices (\emph{e.g.}~\citeNP{speer89}) were not forming
above the MEF during the Flow Mow cruise period.  The similarity of the flow
above the ridge during the cruise to the flow recorded during much longer
periods (\emph{e.g.}~\citeNP{franks92}) suggests that the observed variability
is probably typical.  This indicates the dispersion of MEF plumes is governed
predominantly by multidirectional advection, rather than plume-induced
rotational dynamics (\emph{c.f.}~Section~\ref{sec:plumeform}). 

% Maybe should talk here more about the patterns in the upper level
% hydrography, and defer the following "lessons" to the conclusion chapter??

Each animation of the puff model forced with upper level flow illustrates that
plume distributions are non-uniform and hydrographic variance is high when a
even a single source is venting into multidirectional flow.  Only over long
observation periods should the Endeavour plume be considered a steady state
feature, extending in a downstream direction on average, with concentrations
simply related to plume age or distance from the source.  To properly account
for the complexity of plume distributions, all investigations --- of plume
dynamics, the evolution of conditions with plume age, or the advected heat flux
--- should include observation of the currents.  Successful analysis and
interpretation of the observed plume distributions and variance will hinge upon
the use of advection/diffusion models which provide insights that are otherwise
unavailable.  More sophisticated puff models, for example, could simulate the
dispersion of plumes from multiple sources in non-uniform flow fields, and
thereby help to locate new sources (\emph{e.g.}~\citeNP{lavelle+01}) and
calculate accurate net heat fluxes through a control volume.

Multiple sources distributed along the axial valley contribute to the
hydrographic variability above the Endeavour segment.  Because the mean flow
over the ridge is approximately aligned with the axis, it is reasonable to
expect that plume measurements made above a known source will commonly be
conflated with plumes from nearby sources, especially those that are upstream
with respect to the mean flow.  This possibility is confirmed by the difference
between observations made during the circumnavigation survey and modeled
distributions for the same period.  It is evident that plumes from the High
Rise field were mixing with MEF plumes, a situation which underscores the
importance of assessing $\Delta\theta$ on both up- and downstream surfaces when
measuring net horizontal heat flux through a specified volume.
% Really should look at bks vs delTh on opposite sides during streaming flow to
% see if there is a High Rise signature on one face and the same signature with
% MEF's added on the downstream face!

% Good lead in for discussing past measurements... see notes at end of chapter?
% Or save this juicy stuff for Ch 5?

The multidirectional, oscillatory cross flow also affects the vertical
distribution and variability of plumes.  The rise height $z_r$ of a buoyant
plume in a steady cross flow $U$ is \begin{equation} z_r = 2.6
(\frac{B_o}{UN})^{1/3} \label{crossrise} \end{equation} where $B_o$ is the
source buoyancy flux, $N$ is the buoyancy frequency in the ambient fluid (a
measure of the density stratification), and 2.6 is a empirically derived
constant \cite{briggs69,middleton+thomson86}.  In the ridge crest environment
where cross flows are amplified and of the same or greater magnitude than
vertical velocities in rising plumes, this rise height equation is more
appropriate than the expression derived for quiescent conditions by
\citeN{turner73}: $z_r=5.0 B^{1/4} N^{-3/4}$.  Dividing the derivative of
Equation~\ref{crossrise} by Equation~\ref{crossrise} itself yields an
expression of how a change in rise height $\Delta z_r$ relates to a change in
cross flow velocity $\Delta U$ (J.~W.~Lavelle, pers.\ comm.):
\begin{equation} \frac{\Delta z_r}{z_r} = -\frac{1}{3}\frac{\Delta U}{U}.
\label{dzdu} \end{equation} The observed variations in plume rise height
indicate $\Delta z_r\simeq25$\,m.  With a typical value of $z_r$ above the MEF
(300\,m) and $u$ taken to be either 0.01\,m/s or 0.05\,m/s, Equation~\ref{dzdu}
suggests that $\Delta U$ of only 0.25\,cm/s to 1.25\,cm/s will cause the
observed $\Delta z_r$ of $\sim$25\,m.  If $\Delta U$ is $\sim$0.05\,m/s, a flow
variation that is common in the hourly mean flow $u\simeq0.05$\,m/s over the
Endeavour, Equation~\ref{dzdu} with $z_r\simeq300$\,m yields $\Delta
z_r\simeq$90\,m.  This implies that MEF plumes will be distributed by typical
cross flow over Endeavour in a layer that extends $\pm90$\,m around the average
equilibration depth for any particular source.   The observed thickness of the
plume layer over the MEF is $\sim$200\,m (Figure~\ref{full_waterfall}).


\subsection{Comparison of vertical and horizontal heat flux}

The magnitude of the isohaline heat fluxes $H_v$ and $\overline{H_{qs}}$ should
be comparable, assuming that $H_v$ during the $\sim$24\,hr survey of the side
surfaces of the upper control volume was steady and equal to the mean observed
vertical flux $\overline{H_v}$ \cite{stahr+03}.  
% This is a good place to cite the actual values of Hv for the 2 estimates made
% during ABE50!!!
Table~\ref{uppercomp_table} juxtaposes the means, standard errors, and
uncertainties of horizontal and vertical flux for the upper control volume.
The mean quasi-synoptic heat flux, $\overline{H_{qs}}$ is
$\sim$2/3$\overline{H_v}$; the magnitudes are equivalent, given the standard
errors of the 2 means.  The uncertainty of $\overline{H_{qs}}$ is 2--3$\times$
the uncertainty of $\overline{H_v}$. 

\begin{table}

\caption[Comparison of vertical and horizontal heat flux in the upper control
volume] {Comparison of vertical and horizontal heat fluxes in the upper control
volume measured during the Flow Mow study, and the horizontal flux
associated with the MEF plume measured by \citeN{thomson+92}.  The error
estimate for $\overline{H_v}$ and $\overline{H_{qs}}$ is the standard error of
the mean, while for the instantaneous flux of \citeN{thomson+92} it is
based on the approximate range of magnitudes observed in the entire plume
versus its core.}

\label{uppercomp_table}

\begin{center}
\begin{tabular}{lrrr}
Heat flux                     & Mean     &  Error     & Uncertainty      \\
                              & (MW)     &  ($\pm$MW) & (\%)              \\
\hline
$\overline{H_v}$              & 550      &  100       &  18  \\
$\overline{H_{qs}}$           & 378      &  182       &  47  \\
\hline
\citeN{thomson+92}            & 12000    & 6000       &  50  \\
\end{tabular}
\end{center}
\end{table}

A partial explanation for the underestimation of $H_v$ by $H_{qs}$ 
%and $\overline{H_h}$ 
is the 30--40\,m gap between upper and lower control volumes.  Assuming that
any fluid that escaped assessment had a mean $\Delta\theta$ equal to that in
the upper portion of the plume and was advected with the same velocity that was
acting on the surface above, then the heat flux estimates can be adjusted by
the ratio of the gap height to the surface height, $(Z_{gap}/Z_{surf} = (30$ to
$40)/220 =$14--18\%.  Making an intermediate upward adjustment of 15\% results
in $H_{qs}=$434.
% and $\overline{H_h}=$237.  

The discrepancy between $H_v$ and $H_{qs}$ may also be attributed to: (1) the
non-synoptic observations which underlie $H_{qs}$; (2) variations in currents
assumed to be uniform; (3) heat flux contributions from non-MEF sources not
accounted for perfectly in the non-synoptic observations; and (4) the strong
dependence of the mean value of the instantaneous net horizontal flux on the
observation duration (Figure~\ref{std_dur}), which in the case of the
circumnavigation survey was only $\sim$25\,hr.  The flow during the
circumnavigations, unusual in that oscillations dominated the mean flow, may
also have elevated variance above typical values.  Nevertheless,
$\overline{H_{qs}}$ is equivalent to $\overline{H_v}$, within the standard
error of the means. 

The ``instantaneous'' horizontal heat fluxes estimated by \citeN{thomson+92}
are juxtaposed with $H_{qs}$ in Table~\ref{uppercomp_table} to emphasize
that the 2 estimates may be consistent.  There are 2 explanations for the
factor of 10--20 difference between the 2 estimates.  First, the puff model
indicates that peak instantaneous heat flux through a single surface can be
much higher than the input flux.  This is confirmed by observations: $H_h$
through individual surfaces during the circumnavigation survey were as high as
2000\,MW (Figure~\ref{quasiplot}), almost 4$\times$ the probable input flux,
$H_v=$550\,MW.  It is possible that the ``instantaneous'' estimate of
\citeN{thomson+92} would be greatly reduced to a magnitude closer to $H_{qs}$
if an upstream surface had been assessed and used to compute a \emph{net}
horizontal flux through a control volume enclosing the MEF.  Second, the
modeled time series of instantaneous net flux has peaks of $\sim$5000\,MW,
roughly $10\times$ the 600\,MW input flux.  The modeled peaks are infrequent
enough that the standard deviation during an average 25\,hr period is roughly
an order of magnitude lower than peak values.  This implies that even if an
upstream surface had been assessed and found to have background conditions
($\Delta\theta=$0), the ``instantaneous'' estimate of \citeN{thomson+92},
10--20$\times$ $H_{qs}$ or $H_v$, would not constitute an unreasonable peak
value of the MEF instantaneous net heat flux.
