% Scott's thesis: Chapter 2
 
\chapter{Plumes and Heat Flux in the Valley -- Rectilinear Flow}
\label{ch:lower}

During the Flow Mow study, currents within the axial valley near the Main
Endeavour field consisted of a northward $\sim$1-5~cm/s mean flow superimposed
on along-axis tidal oscillations of comparable amplitude.  When this type of
flow distributes energy from a hydrothermal source within a control volume, the
net outward heat flux through the bounding control surfaces can vary
significantly over time.  Even if the source heat flux is constant, temperature
anomalies in the water column can change quickly as thermal energy pools above
a vent during slack flow periods and streams away from it during peak cross
flow.  Attempts to measure the total flux through control volume surfaces can
result in a time series that has surprisingly high variance and, depending on
how temperature and velocity signals are averaged, a mean magnitude
dramatically different from the source magnitude \cite{wetzler+98}.  

In this chapter I begin with a description of the method used to calculate
total lateral heat flux from the MEF within $\sim$100\,mab.   After presenting
the temperature anomaly measurements made on or near the vertical control
surfaces, I combine the temperature and current data to estimate the net
horizontal heat flux through the side surfaces of the lower control volume
(Figure~\ref{fmcv}).  Subsequently, I use a simple advection-diffusion model to
interpret the observed flux magnitudes and gain insight into the flux variance
caused by variable currents near the MEF.  Finally, I combine the horizontal
and vertical heat flux estimates in the lower control volume to infer the
partitioning of heat flux between diffuse and focused hydrothermal venting
within the MEF.


\section{Calculated horizontal heat flux}

Heat flux is calculated by integrating over a control volume surface the
product of $\Delta \theta$ and current velocity.  The net horizontal heat flux
$H_h$ out of the control volume is the sum, minding flux direction, through all
4 vertical control surfaces (\emph{c.f.}~Equation~\ref{lowerbudgetu0}).  In
this section, however, I make 2 assumptions that simplify the heat flux
calculation, both based on the nature of the flow within the axial valley and
the observational limitations of the Flow Mow data set.

First, I assume the consistent northward mean flow and rectilinear tidal
oscillations within the axial valley (\emph{c.f.}~Section~\ref{sec:flow}) mean
that only the north and south control surfaces are important in the computation
of net horizontal flux from the MEF.  That is, I assume the heat flux through
east and west surfaces is negligible.  This is reasonable because the east and
west surfaces are oriented parallel to the ridge axis and are therefore
approximately aligned with the local, rectilinear flow.  This assumption is
also necessary because the Flow Mow hydrographic program, motivated by previous
observation of rectified, along-axis flow at the MEF, focused on monitoring the
north and south surfaces repetitively and at multiple tidal phases, while the
east and west surfaces were surveyed only 1--2 times.

Second, I assume that measurements from current meters moored $\sim$1.1\,km north
or south of the MEF can be used to calculate accurately the horizontal heat
flux at the MEF control surface locations.  This is justified by the northward mean flow within the axial
valley consistently observed at meters north and south of the MEF (FM-N15 and
FM-S50, Table~\ref{cm_table}), and by the spatial coherence of the oscillatory
flow between this pair of meters.  The assumption is also necessary because
velocity measurements were not made alongside the CTD observations on the
lowered instrument package.  

In this section, with these simplifications, I calculate a range of observed
net horizontal heat fluxes from the MEF using temperature anomaly data from the
north and south control surfaces and northward mean velocities of 1--5\,cm/s
measured at the moorings.  The first consideration, tackled in the next
subsection, is making an accurate assessment of $\Delta\theta$ on the control
surfaces.  The subsequent subsection addresses integration of the
$\Delta\theta$ and current meter observations to calculate net horizontal heat
flux.  Estimates of vertical heat flux through the upper surface are presented
in \citeN{stahr+03}.  


%A fundamental assumption in the Flow Mow methodology was that hydrographic
%conditions near the MEF control surfaces would be uniform.  In a steady flow
%field, thermal homogeneity on the scale of the MEF would guarantee that
%advection of heat in one side of the field would be balanced by advection out
%the opposite side.  Additionally, any heat entrained by plumes rising within
%the control volume could be accounted for by subtracting the mean
%$\Delta\theta$ observed on the perimeter from that observed on the top.

%Contrary to our expectations, hydrographic surveys in the MEF and within
%$\pm$500~m along-axis show that near-bottom variability is high.  Upon spatial
%and temporal averaging, however, an overall pattern emerges: near the bottom
%it is warmer north of the MEF than south of it.

\subsection{Observed $\Delta\theta$}
\label{sec:mef_dth_lower}

During the Flow Mow field program, spatial hydrographic variability was
monitored on the side surfaces of the lower control volume (Figure~\ref{fmcv})
through repetitive surveys with ABE or a navigated CTD.  Transects of
individual surfaces took $\sim$1.5--3\,hrs, came within $\sim$5\,m of the sea
floor, and reached maximum heights of $\sim$100\,mab.  ABE surveys, primarily
in the form of descending sequences of lateral transects, acquired hydrographic
data with horizontal resolution as high as $\sim$2~samles/m and vertical
resolution of $\sim$1~sample/10\,m (Figure~\ref{ns_abe_walls}).  Vertically
oscillating CTD tows (VOTs) along side surfaces generated data with vertical
resolution as high as $\sim$4~samples/m and horizontal resolution of
$\sim$1~sample/25\,m (Figure~\ref{allctd_NSwalls}).  Temporal hydrographic
variability was also monitored during the field program, most intensely during
10--14\,hr-long vertically oscillating casts (VOCs) between 2100 and 2200\,m
(Figure~\ref{nosomef_waterfall}) at fixed locations $\sim$500\,m north and
south of the MEF.  

Samples of $\theta$ and $S$ were converted into $\Delta\theta$ (isohaline anomaly)
using an expression like Equation~\ref{sanom}, but with a polynomial fit to the
average background $\theta$--$S$ relationship observed at CTD station~1~and~2
between 1700--2375\,m: $\Delta\theta=\theta-(-24480.315+1409.267 S-20.283
S^2)$.  In preparing the figures in this section, the location of navigated
$\Delta\theta$ values from ABE or CTD were projected orthogonally onto the
appropriate side surfaces of the lower control volume.  This was necessary
because most ABE and CTD surveys extended above or beyond the edges of the
control surfaces; only those measurements located on the control surfaces after
orthogonal projection were included in heat flux estimations.  Typical lateral
separation between a point projected on a control surface and the corresponding
ABE or CTD measurement location was $<20$\,m.  The projected $\Delta\theta$
values on each surface were then aggregated into bins of height and width,
averaged, and color-coded according to the resultant mean $\Delta\theta$ value.
Bin size was specified with the goal of containing data from adjacent tracks.
For ABE surveys, bins with dimensions of $\Delta x=$10~m and $\Delta z=$25~m
typically contained $\gtrsim$100 samples/bin; for VOTs, $\Delta x=$100~m,
$\Delta z=$5~m, and typical bins held $\gtrsim$40 samples/bin.  Bins with no
data were left blank (white).  Using this bin-averaging technique, no
contouring was performed.  

In this subsection, I present 3 perspectives on $\Delta\theta$ variability and
patterns near the MEF within the axial valley ($<2100$\,m).  First, I examine
spatial variability with the ABE surveys of the north and south surfaces of the
lower Flow Mow control volume.  A second, similar description of spatial
variability comes from CTD surveys of the same surfaces.  Finally, I illustrate
temporal variability with 2 time series acquired at CTD stations $\sim$500\,m
north and south of the MEF boundaries.

\subsubsection{ABE surveys of north and south surfaces}

The $\Delta\theta$ fields on the north and south surfaces of the MEF control
volume were observed during 3 separate ABE dives (45, 46, and 50) and are
presented in Figure~\ref{ns_abe_walls}.  Each of the 6~panels in the figure
represents a separate transect of a surface with $\Delta\theta$ values overlain
by the ABE track line.  Dives 45 and 50 have mean anomalies on their north
surfaces that are 0.013--0.020$^\circ$C larger than on their south surfaces.
Dive 46 has a mean anomaly on its south surface that is 0.008$^\circ$C larger
than on its north surface; even when only the upper 50~m of the
unequally-covered surfaces are compared for Dive~46, the mean anomaly at the
south surface (0.077$^\circ$C) is slightly larger than at the north surface
(0.074$^\circ$C).  Successive north surfaces were separated by 13.8 and
199.9\,hr, and south surveys were separated by 23.0 and 122.1\,hr.  The results
thus show the strong variations in time of excess heat at the north and south
surfaces.

\begin{figure}[htbp] \begin{center}
\includegraphics[width=0.8\textwidth]{figs/fm1/allabe_NSwalls.eps2}
\end{center} \caption[ABE survey of $\Delta\theta$ on surfaces north and south
of MEF]{$\Delta\theta$ on south and north control surfaces during ABE dives 45,
46, and 50.  Successive north surfaces (right 3 panels) were separated by 13.8
and 199.9\,hr, and successive south surveys (left 3 panels) were separated by
23.0 and 122.1\,hr.  The title over each panel contains: a number indicating
the order in which the surfaces were surveyed; the dive number; and the mean
$\Delta\theta$ of all bins (noted parenthetically).  Averaging bins are 10~m
wide and 25~m high.  White areas have no data.  Annotations in lower left
corner of each panel are times in hours: first the survey duration for that
surface, then the time since the end of the previous surface (\emph{e.g.}
survey 5:N began 115.7\,hr after the end of survey 4:S).  Black line represents
the ABE track.  Varying maximum survey depths are due to changes in ABE's
bottom-avoidance algorithm.} \label{ns_abe_walls} \end{figure}

Figure~\ref{ns_abe_walls} also shows the degree of spatial variability in plumes in
this depth range and at this distance from the MEF sources.  On any given pass
across the 300~m wide surface, plumes or thermal patches as narrow as 20~m are
observed.  Sequential passes, sometimes separated only 10~m vertically and
$\sim$10-20~min temporally, commonly encounter $\Delta\theta$ that varies by as
much as a factor of 2.  

\subsubsection{CTD surveys of north and south surfaces}

An additional perspective on the $\Delta\theta$ spatial variability comes from
data taken on 3~VOTs (Figure~\ref{allctd_NSwalls}).  During station~10, the CTD
was towed back and forth across the north surface 3 times, generating 3 surveys
of the north surface.  The south surface was similarly surveyed $\sim$5.5 times
during stations~12~and~13.  Since the VOTs were continuous, there is no
temporal separation between sequential surveys of a given surface; when the end
of a surface was reached during a particular station, the ship immediately
reversed direction and began another survey of the same surface.  Each surface
survey took $\sim$80--100\,min and extended to within 5--10\,m of the sea
floor. 

When all samples from 2100--2200\,m are averaged together on each surface in
Figure~\ref{allctd_NSwalls}, north surfaces have mean $\Delta\theta$ from
0.053--0.091$^\circ$C, while south surfaces have mean $\Delta\theta$ from
0.053--0.076$^\circ$C.  In the \emph{sequential} panels of
Figure~\ref{allctd_NSwalls} (Stn~10: N\#1--3, Stn~12: S\#5, Stn~13: S\#1), the
first north surface has the highest mean $\Delta\theta$, but many of the south
surfaces have greater mean $\Delta\theta$ than the second and third north
surfaces.  Compared to north surfaces, the south surfaces are more uniformly
warm in their upper 25~m.  The lowest 25-50~m of the south surfaces are
variable and generally cooler than in the same depth range on the north
surface.  

When the surfaces in Figure~\ref{allctd_NSwalls} are sorted into north and
south groups and averaged, the mean $\Delta\theta$ on the south surface
(0.0677$^\circ$C) is slightly less than on the north surface (0.0690$^\circ$C).
This result is consistent with the overall average $\Delta\theta$ differences
detected in the ABE survey of the north and south surfaces.  Though the
difference is smaller in this case, it is well within the the sensor
resolutions  (\emph{c.f.}~Section~\ref{sec:instrumentation}).

\begin{figure}[htbp] \begin{center}
\includegraphics[width=0.8\textwidth]{figs/fm1/allctd_NSwalls.eps2}
\end{center} \caption[CTD survey of $\Delta\theta$ on surfaces north and south
of MEF]{Isohaline potential temperature anomaly $\Delta\theta$ on south (left
2~columns) and north (right column) control surfaces during CTD stations 10,
12, and 13.  Title above each panel indicates: CTD station number; south (S) or
north (N) surface; order (\#) in sequence of surface surveys during each
station; mean of all bins (noted parenthetically).  Station 12 began
$\sim$3.8~hr after station 10 ended, and station 13 began $\sim$14.2~hr after
the end of station 12; sequential surveys during each station are temporally
contiguous.  Bottoms of successive vertical oscillations are spaced
$\sim$20~min apart; surfaces were surveyed in $\sim$80--100~min.  Black line
represents CTD track.  Averaging bins are 100~m wide and 5~m high.  White areas
have no data.  } \label{allctd_NSwalls} \end{figure}

Interestingly, in almost every survey of the south surface that approached
within 50~mab (Figures~\ref{ns_abe_walls}~and~\ref{allctd_NSwalls}), a 20--50~m
wide and $\lesssim$25~m tall patch of fluid with
$\Delta\theta\sim$0.06--0.09$^\circ$C was encountered within 25~mab, usually
near $X=200$\,m.  It is possible that the patch was a plume from Quebec
(\emph{c.f.}~Section~\ref{sec:mefhta}), a diffuse flow site located
$\sim$250\,m south of the MEF (Figure~\ref{quebec}), that was advected
along-axis in northward mean flow.  The rise height of the Quebec plume cannot
be constrained because the salinity and volume flux of the fluid venting at
Quebec are unknown.  During the Flow Mow study, however, the Quebec plume was
detected at depths of 2125--2200\,m (Figure~\ref{quebec}) in 2~VOTs that
criss-crossed and circumnavigated the 1995 Alvin coordinates where Quebec was
discovered.  The peak $\Delta\theta$ sensed in the Quebec vicinity was
0.12$^\circ$C and 10~m thick layers had $\Delta\theta$ of 0.05--0.09$^\circ$C.
These values are only slightly higher than the mean $\Delta\theta$ values of
the warm patch on the south control surface.  Assuming a small source size
($\lesssim10$\,m) and reasonable value for near-bottom diffusivity
(0.4\,m$^2$/s, \citeNP{okubo71}), diffusion of a plume from Quebec during
advection at 1--5\,cm/s over the 250\,m distance to the south surface would result
in a plume width of roughly 60--140\,m \cite{csanady73}.  This width is also
consistent with that of the patches observed on the south surface.

\begin{sidewaysfigure} \begin{center}
\includegraphics*[width=.45\textwidth]{figs/fm1/quebec/stn21_25.dTh_map.eps2}\hfill
\includegraphics*[width=.45\textwidth]{figs/fm1/quebec/stn21_25.th-z.eps2} 
%\includegraphics*[width=.45\textwidth]{figs/fm1/quebec/stn21_25_bkg.th-S.eps2}
\caption[Survey of diffuse flow site Quebec: $\Delta\theta$ map and
profiles]{Map on left shows the CTD track during 2 VOTs around and over the
Quebec diffuse flow site (47$^\circ$56.62'N, 129$^\circ$6.0'W).  Overlain in color are ranges of $\Delta\theta$
encountered along the tracks; see legend for temperature ranges.  Quebec
coordinates, based on a 1995 Alvin dive, are delineated by the small box
containing the $+$ symbol.  The southern end of the MEF is near the top of the
map, along with the beginning of the track for CTD station 20, which traversed
the east side of the MEF.  The $\Delta\theta$ profiles at right are derived
from the 2 stations that surveyed Quebec (21 and 25).  } \label{quebec}
\end{center} \end{sidewaysfigure}

A probable explanation for the temporal variability of average $\Delta\theta$
on individual surveys of the north and south control surfaces is advection of
MEF plumes by oscillatory currents and mean flows within and above the axial
valley.  This possibility is further explored through analysis of time series
from stationary CTD profiles.

%% NEED to address idea that mean pattern of north warm (due to mean flow)
%holds, even though variability is high (and expected to be high) due to
%oscillations.
%
%  USE bow tie plots!!!

\subsubsection{CTD time series north and south of the MEF}

The mean northward flow that had been indicated by the 1995 current meter
motivated special emphasis during the Flow Mow field program on monitoring and
comparing 2 CTD survey areas (dubbed ``NoMEF'' and ``SoMEF'') located
$\sim$500~m along-axis north and south of the MEF boundary
(Figure~\ref{abebathy}).  Both areas were monitored at various tidal phases for
periods of 0.5--14\,hr during the CTD stations spaced 1--5\,days apart.  NoMEF
was assessed during stations~3, 5, 14, and 28, while SoMEF was studied during
stations~4, 6, 7, and 33 (Figure~\ref{nosomef_waterfall}).  During stations 28
and 33, NoMEF and SoMEF were reoccupied for 10 and 14\,hrs, respectively,
approximately a full semidurnal tidal period.  NoMEF was monitored during
station 28 for about 10\,hr during a VOC between 1800 and 2200~m.  3\,days
later during station 33, SoMEF was assessed ($\sim$200~m west of previous SoMEF
CTD stations 4 and 6) for 14\,hr during a VOC between 1800 and 2150~m.  To
examine the temporal variability of $\Delta\theta$, samples from each CTD
station are averaged in depth-time bins with dimensions chosen to contain at
least 2 sequential CTD casts and $\sim$20 samples ($\Delta z=$5~m, $\Delta
t=$30~min).  Then, the intermittent surveys are concatenated, creating a series
of observations characterizing the NoMEF and SoMEF areas
(Figure~\ref{nosomef_waterfall}).

\begin{figure}[thbp] \begin{center}
\includegraphics[width=0.8\textwidth]{figs/fm1/Dth.nosomef.image.alldeep.eps}
\end{center} \caption[Time series of $\Delta\theta$ in the NoMEF and SoMEF
areas (2100--2200~m)]{Time series $\Delta\theta$ from vertical CTD casts in the
NoMEF area (top panel) and SoMEF area (bottom panel).  Before plotting, data
were averaged in vertical bins 5~m high and $1/2$\,hr wide.  White areas
contain no data.  Dotted black line represents CTD depth history.  Vertical
black lines mark temporal breaks of $\sim$1-5~days between stations, each of
which is numbered in lower left corner.  Station 33 does not extend below
2155~m because it was located west of stations 4, 6, and 7, where the western
scarp shoals to $\sim$2160~m (see dual SoMEF locations,
Figure~\ref{location-map}).  For additional context, see
Figure~\ref{full_waterfall}.} \label{nosomef_waterfall} \end{figure}

The $\Delta\theta$ profile at NoMEF and SoMEF changes with time
(Figure~\ref{nosomef_waterfall}).  At NoMEF $\Delta\theta$ is elevated in an
upper layer (2100--2150\,m) relative to the layer within 50\,mab.  In both
depth ranges $\Delta\theta$ is consistent from hour to hour, but changes by
$\sim0.05^\circ$C over time scales longer than a semi-diurnal half-period.  An
upper layer with relatively high $\Delta\theta$ is also present at SoMEF during
stations~4, 6, and 7; during the same period $\Delta\theta$ in the SoMEF lower
layer varies only $\sim$0.02$^\circ$C, less than the variations in the lower
layer at NoMEF.  While large near-bottom $\Delta\theta$ attributed to a plume
from Quebec was observed on south control surfaces
(Figures~\ref{ns_abe_walls}~and~\ref{allctd_NSwalls}), no similar anomalies are
observed at SoMEF, which is located south of Quebec (Figure~\ref{abebathy}) and
therefore upstream with respect to the northward mean flow in the axial valley.
The warmer upper layer evident in most of Figure~\ref{nosomef_waterfall} was
not present during station~33 (which was located further east and up on the
western scarp of the axial valley, Figure~\ref{abebathy}), but $\Delta\theta$
still varied during station~33 up to $\sim$0.04$^\circ$C at some depths over
time scales of about a semi-diurnal half-period.  These observations suggest
the $\Delta\theta$ field at this distance from the MEF changes over
semi-diurnal time scales, while closer to the MEF sources the $\Delta\theta$
distribution on particular control surfaces in Figure~\ref{allctd_NSwalls}
changes between CTD surveys, on time scales of 1--2\,hrs.   Thus, temporal
variability appears to increase with proximity to the MEF. 

%Within the bottom 50\,m, the variance of $\Delta\theta$ also seems to increase
%with proximity to hydrothermal sources.  This is evident in the extreme values
%of $\Delta\theta$ at any particular depth in Figure~\ref{nosomef_waterfall}
%(with those in Figures~\ref{ns_abe_walls}~or~\ref{allctd_NSwalls}.

Despite the exhibited temporal variability, the average $\Delta\theta$ taken
over all depths and times (Figure~\ref{nosomef_waterfall}) is 0.050$^\circ$C at
NoMEF and 0.043$^\circ$C at SoMEF.  As was the case with the control surface
surveys, the \emph{spatial and temporal average} $\Delta\theta$ is greater
north of the MEF than south of it.

\subsubsection{Synopsis of spatial and temporal variability}

A final perspective on axial valley hydrography comes from a nearly synoptic
observation of $\Delta\theta$ on \emph{all 4} MEF control surfaces,
accomplished during ABE dive 50 in 15.4\,hr (Figure~\ref{abe50all}).  ABE
surveyed the north surface first, taking 1.8\,hr.  After completing a survey of
the top surface in 6.3~hr \cite{stahr+03}, ABE surveyed the south, east, and
west surfaces sequentially, taking 1.0, 4.2, and 1.1~hr on each surface,
respectively.  The transition between south and east surfaces took 18~min,
while the last transition, from east to west surfaces took 40~min.  

Figure~\ref{abe50all} shows that fluid with the smallest $\Delta\theta$ is
predominantly near the bottom, and on the east and southeast sides of the MEF.
This is consistent with other casts that contain relatively rare encounters
with near-zero $\Delta\theta$ axial valley fluid (\emph{c.f.}~stations~20, 21,
and 25 in Figure~\ref{quebec}); the deep areas to the south and east of the MEF
tend to have the smallest $\Delta\theta$.  This pattern supports the idea that
\emph{on average} the mean northward flows bring relatively cool fluid into the
MEF through the south control surfaces and transport relatively warm
hydrothermal plumes through the north surfaces.   

\begin{figure} \begin{center}
\includegraphics[width=.75\textwidth]{figs/fm1/abe50allsides.eps}
\caption[Near-synoptic $\Delta\theta$ distribution on MEF
perimeter]{$\Delta\theta$ distribution on all 4 side surfaces of the MEF
control volume, acquired within a 15.4~hr period during ABE dive 50.
Observations are projected on to vertical planes aligned with the mean track of
ABE on each side, rather than on to the control surfaces.  Coordinates are
relative to the Alvin origin.}  \label{abe50all} \end{center} \end{figure}

Considered along with the other Flow Mow observations presented in this
section, Figure~\ref{abe50all} also describes a high level of $\Delta\theta$
variability on the MEF control surfaces.  While it is not possible to
distinguish between spatial and temporal variability with a moving sensor, the
successive ABE or CTD transects of the control surfaces imply scales of
$\lesssim$10\,m or $\lesssim$20\,min.  On the whole, the Flow Mow observations
evidence a level of hydrographic heterogeneity, or ``patchiness,'' within the
Endeavour axial valley that was poorly characterized before the Flow Mow study.
Past hydrographic surveys of the MEF vicinity suggested that $\Delta\theta$
within the valley was generally homogeneous, despite the rare serendipitous
encounters with unusually high $\Delta\theta$ fluid that sometimes led to
successful inference of diffuse vent locations \cite{veirs+99}.  Previous
investigations generally viewed the bottom 50--100\,m as a relatively
well-mixed boundary layer, vertically and horizontally homogeneous, and less
anomalously warm than the overlying plume layers
(\emph{e.g.}~\citeNP{thomson+89}).  

Calculation of the horizontal heat flux through the lower control volume
requires the mean isohaline potential temperature anomaly (hereafter
$\overline{\Delta\theta}$) observed on the north ($\overline{\Delta\theta}_N$)
and south ($\overline{\Delta\theta}_S$) control surfaces.  Later in this
chapter, an advection/diffusion model forced by tidal oscillations is used to
understand the variability of $\overline{\Delta\theta}$ observed near the MEF.
Here, however, I obtain an estimate of $\overline{\Delta\theta}_N$ and
$\overline{\Delta\theta}_S$ by generating depth-binned histograms of
$\Delta\theta$ samples acquired north or south of the MEF --- either during ABE
or CTD surveys of the control surfaces, or CTD surveys of the NoMEF and SoMEF
areas.

Observations of $\Delta\theta$ at MEF both above and below the ridge include
those data in Figures~\ref{ns_abe_walls}, \ref{allctd_NSwalls}, and
\ref{nosomef_waterfall}, and all data acquired \emph{above} the ridge crests
during the same set of stations (to be further analyzed in
Chapter~\ref{ch:upper}).  Aggregating all these data into north and south groups
and sorting into 25~m depth bins, histograms of $\Delta\theta$ are generated
for each bin.  The resulting distributions of $\Delta\theta$
(Figure~\ref{depth-hist}) characterize the full range of depths (1800--2200~m)
influenced by plumes in the vicinities north and south of MEF, and therefore
provide some context for Figures~\ref{ns_abe_walls}, \ref{allctd_NSwalls}, and
\ref{nosomef_waterfall} which portray data from only the deepest 4 bins
(2100--2200\,m).  While the spatial and temporal variability of near-bottom
$\Delta\theta$ around the MEF was higher than expected, the $\Delta\theta$
variance is low in the bottom 100~m relative to the depths above 2100~m, where
cross-axis flow increases the variability of plume distributions
(c.f.~Section~\ref{sec:ns_surfs}).  

\begin{figure}[htbp] \begin{center}
\includegraphics[width=0.6\textwidth]{figs/fm1/all_zbin_dth_hist_nomef.eps2}
\includegraphics[width=0.6\textwidth]{figs/fm1/all_zbin_dth_hist_somef.eps2}
\end{center} \caption[Histograms of depth-binned $\Delta\theta$ observed north and south
of MEF]{Stacked histograms of depth-binned $\Delta\theta$ observed north
(upper) and south (lower) of the MEF.  Depth range of each bin is 25~m and
upper extent of each bin is noted on the left axis.  For reference, the nearby
ridge crests have a mean depth of $\sim$2100--2125~m.  $\Delta\theta$ for each
distribution is tabulated at right.  North data is from CTD stations 3, 5, 10,
14, and 28, and ABE dives 45, 46, and 50; south data is from CTD stations 4, 6,
7, 12, 13, and 33, and ABE dives 45, 46, 50.} \label{depth-hist} \end{figure}

Figure~\ref{depth-hist} can also be used to quantify north--south differences
in $\overline{\Delta\theta}$.  In the bottom 100~m, each 25~m bin has a
$\overline{\Delta\theta}$ that is greater in the north.  The average
$\overline{\Delta\theta}$ taken over these 4 deepest bins is 0.063$^\circ$C in
the north, versus 0.056$^\circ$C in the south.  These are the values I use to
estimate horizontal heat flux in the next section.


\subsection{Horizontal heat flux estimation}

The patchiness of hydrothermal plumes in the axial valley indicates thermal
heterogeneity and suggests that unequal lateral fluxes through the different
control surfaces are probably commonplace.  In light of this variability, I use
$\overline{\Delta\theta}$ from the bottom 100~m of Figure~\ref{depth-hist} to
characterize the temperature field on the north and south surfaces of the lower
Flow Mow control volume (Figure~\ref{fmcv}).  The mean net horizontal heat flux
$H_h$ is then computed as the sum of the \emph{mean} horizontal heat flux
through the north surface ($\overline{H}_N$) and south surface
($\overline{H}_S$) via an adaptation of Equation~\ref{lowerbudget}:
\begin{equation} 
H_h=\overline{H}_N+\overline{H}_S = \rho c_p \overline{v} A (\overline{\Delta\theta}_N - \overline{\Delta\theta}_S).
\label{hh}
\end{equation} {
In this expression, the area $A$ ($3\times10^4$\,m$^2$) is assumed to be the
same for both north and south surfaces.  Additionally, the along-axis component
of mean flow $\overline{v}$ is assumed to be identical at both surfaces.  The
product of the reference density $\rho$ and heat capacity $c_p$ is taken to be
4.2\,MJ$\cdot$m$^{-3}\cdot^\circ$C$^{-1}$.

An important difference between Equation~\ref{hh} and
Equation~\ref{lowerbudget} is that here $\overline{\Delta\theta}$ is used in
place of $\overline{\theta}$.  This means that $\overline{H}_N$ and
$\overline{H}_S$ are isohaline heat fluxes, the horizontal components of the
$H_{isoS}$ term in Equation~\ref{isohalineheat}.  Since progressive vector
diagrams show that $v$ consistently dominates the across-axis component near
the MEF (Figure~\ref{valley_pvds}), heat fluxes through the east and west
control surfaces are assumed negligible.  Consequently, the \emph{net}
horizontal isohaline heat flux $H_h$ is the sum of $\overline{H}_N$ and
$\overline{H}_S$.  $H_h$ represents the heat flux from MEF sources that leaves
the control volume by horizontal advection within 100~m of the sea floor.  The
remaining MEF source heat flux passes vertically through the top control
surface.  

Using the north--south difference in $\overline{\Delta\theta}$ values from the
bottom 100~m of Figure~\ref{depth-hist} ($0.063-0.056=0.007^\circ$C) and the
values of $\overline{v}$ observed north and south of the MEF ($\sim$1\,cm/s at
FM-S50 and $\sim$5~cm/s at FM-N15), I obtain an estimate of $H_h$: 10--50\,MW.
Changing the difference $\overline{\Delta\theta}_N-\overline{\Delta\theta}_S$
by 0.001$^\circ$C changes the value of $H_h$ by 1.25\,MW, while increasing the
magnitude of $\overline{v}$ by 1~cm/s increases $H_h$ by 10\,MW.

The Flow Mow hydrographic survey resolved the MEF temperature field at
unusually high temporal and spatial scales.  Despite the thermal variability,
the north and south control surfaces were sampled frequently enough, and at a
wide enough range of tidal phases, to enable estimation of the \emph{mean} net
flux, $H_h$.  However, the observations are too sparse to capture the
variations of \emph{instantaneous} net flux out of the control volume.  Indeed,
given the limitations of the acoustic navigation network and our instruments,
it was a challenge to assess a single side and a top simultaneously.  It was
impossible to monitor all 4 sides and the top at once, though that ability is
critical if instantaneous fluxes are sought in such variable hydrography.
Consequently, I use synthetic data to understand the high variability and
interpret our sparse, non-synoptic observations.

\section{Modeled horizontal heat flux}

To understand the observed hydrographic variability near the MEF and the effect
of variable currents on heat flux, I model the 2-dimensional distribution of
tracer from a point source subjected to a combination of advection and
diffusion.  The model is similar to advection/diffusion models that have been
used extensively to investigate dispersion in the atmosphere
(\emph{e.g.}~\citeNP{pasquill74}) and ocean
(\emph{e.g.}~\citeNP{csanady73}).  In particular, I use a form of
advection/diffusion model common in studies of pollutant transport and known as
a ``puff'' model \cite{schnell+dey00}.

This section begins with a description of the puff model.  The model is then
used to simulate the dispersion of a plume from the MEF in combinations of
oscillatory and mean flow characteristic of the axial valley.  The main product
of the simulations is a statistical analysis of the horizontal heat flux
through the MEF perimeter, and its relationship to the steady source heat flux.
Finally, the observed horizontal heat flux is discussed with respect to the
model results.

\subsection{The puff model}
\label{sec:puffmodel}

At each time step of a puff model a ``puff'' of tracer is added at the source,
constituting a constant flux into the model domain.   The distribution of
tracer in each puff is governed by the sum of 2 error functions; initially a
top hat profile with half width $r$, the distribution quickly becomes Gaussian.
As each puff ages and is advected according to the current record, its Gaussian
distribution broadens (Figure~\ref{initpuff}).  Thus, each puff diffuses as it
is advected, and the sum of all puff concentrations simulates the plume
distribution.  Similar ``puff'' models have been used to understand atmospheric
plume distributions (e.g.~\citeNP{rao+89}) and more recently to interpret
temperature and current records from moorings near a hydrothermal field
\cite{wetzler+98,lavelle+01}.  (See related \href{http://econscience.org/scott/pubs/thesis/diffuse_endeavour.html}{animations of plume dispersion from
a diffuse hydrothermal vent},
courtesy of J.W.~Lavelle from NOAA's Pacific Marine Environmental Laboratory.)
%\href{http://www.pmel.noaa.gov/}{Bill Lavelle's web site}:
%\\http://www.pmel.noaa.gov/$\sim$lavelle/Review/Workstation/Anim/d6.fli).  

\begin{figure} \begin{center}
\includegraphics*[width=.75\textwidth]{figs/puff/initpuff_curves.eps}
\caption[Concentration curves in ``puff'' model]{Examples of concentration $c$
profiles for an individual puff as a function of distance $x$.  The
distribution in each dimension is governed by the sum of 2 error functions; the
sum of the 2 error functions that intercept $c=\pm1$ describes the initial top
hat profile, but as the error functions evolve with time, their sum yields
increasingly broad Gaussian curves.  In this example, the initial half-width
$r=$15\,m.} \label{initpuff} \end{center} \end{figure}

The puff model domain is 4$\times$4\,km square with a horizontal grid
resolution of 50\,m, resulting in an 80$\times$80 cell grid.  The origin of an
$X$--$Y$ coordinate system is located at the center of the domain.  A single
source is placed at the origin and is assigned a steady heat flux.  The heat
flux, time step, and initial puff dimensions determine the initial temperature
anomaly ($\Delta\theta_o$) of the puff.  For example, a constant source flux of
300\,MW delivered during a $1/2$\,hr time step to an initial (cubical) puff
100\,m on a side ($r=$50\,m) results in an initial uniform temperature anomaly
$\Delta\theta\sim0.1^\circ$C.  Vertical diffusivity is set to zero, so
the puff thickness remains constant throughout its lifetime.  Horizontal
diffusivity is uniform and given a value typical of the interior ocean,
$k=k_x=k_y=0.4$~m$^2$/s \cite{okubo71}.  For computational efficiency, puffs
are eliminated if they move beyond the 4$\times$4\,km domain.  To minimize
rendering time and file size, only a 2$\times$2\,km sub-domain is displayed in
the model animations (Figure~\ref{puff-model}).  

In the first time step, the first puff is emitted at the source.  During the
next time step, it diffuses and is advected according to the spatially uniform
velocity series $\mathbf{v}(t)$, reaching a new position ($x_p$,$y_p$) just as
a new puff is emitted at the origin.  At the end of each time step,  any puff
that moves beyond the 4$\times$4\,km domain are eliminated.  Finally, with $n$
puffs in their new positions, the $\Delta\theta$ contribution from each puff in
the domain (at $x_p$,$y_p$) to each cell in the grid (at $x$,$y$) is computed
to yield the $\Delta\theta$ distribution over the whole domain, an evolving
function of time $t$:
\begin{equation}
\Delta\theta_{x,y}(t)= \sum_{i=1}^{n} 
{ \frac{\Delta\theta_o}{4} 
  \left( \mathrm{erf}( \frac{r+\dif x_i}{\sqrt{4 k_x t_i}} )+\mathrm{erf}(\frac{r-\dif x_i}{\sqrt{4 k_x t_i}} )\right)
  \left( \mathrm{erf}( \frac{r+\dif y_i}{\sqrt{4 k_y t_i}} )+\mathrm{erf}(\frac{r-\dif y_i}{\sqrt{4 k_y t_i}} )\right)  
}
\label{puffsum}
\end{equation}
in which $t_i$ is the age of the $i$th puff, $r$ is the initial puff
half-width, $\Delta\theta_o$ is the initial puff temperature anomly, $k$ is
diffusivity, and $\dif x_i$,$\dif y_i$ are distances from the center of each
grid cell ($x$,$y$) to the center of each puff ($x_p$,$y_p$).  After the
summation is conducted for every grid cell in the domain, the $\Delta\theta$
field is visualized by coloring each cell (according to a palette that does not
change between animation frames).  In each subsequent time step, the value of
$\Delta\theta_{x,y}$ changes when advection alters the spatial distribution of
puffs and diffusion shifts the concentration contributions of individual puffs.

The instantaneous horizontal heat flux through a particular surface of the MEF
control volume is found at each time step by multiplying $\rho$, $c_p$,
$\Delta\theta$, the orthogonal component of flow, the grid cell width, and the
initial puff height for each cell, and summing the products over all cells that
make up the surface.  Alternatively, the time series of $\Delta\theta$ at a
particular point can be recorded and later used to compute mean $\Delta\theta$
and/or heat fluxes.  Note that the summation in Equation~\ref{puffsum} can be
undertaken for any $x$,$y$ position in the domain, not just the centers of the
grid cells.

\subsection{Modeled variance and mean magnitudes}

The horizontal heat flux through the MEF control volume is studied by running
the puff model with observed currents, specifically characteristic oscillations
derived through harmonic analysis of the along-axis component of flow measured
at the 1995 near-bottom current record (MZ25, Table~\ref{cm_table}).  Mean flow
of 0--7\,cm/s is added in steps of 0.5\,cm/s to the oscillatory flow in a
series of simulations.  Each simulation generates a unique $\Delta\theta$ field
that is monitored on 4~control surfaces bounding the MEF, yielding a time
series of net heat flux for each mean flow.  Since the current is aligned with
the east and west control surfaces in this case, only the fluxes through the
north and south surfaces contribute to the net flux.  These simulations reveal
how flux magnitude and variance evolve as the mean flow balances and then
exceeds the amplitude of the oscillations.  

\begin{figure}[htbp] \begin{center}
\includegraphics[width=0.8\textwidth]{figs/puff/puffframe.eps} \end{center}
\caption[Puff model forced by idealized flow]{A single frame from a puff model
animation showing a plan view of a thermal plume extending northward from the
MEF (dashed black rectangle).  In this snapshot, 240\,hr into a 360\,hr
simulation, the puffs are advected by idealized flow (a 1\,cm/s mean added to
an oscillation derived by H.~Mofjeld through harmonic analysis of the record
from the MZ25 current meter; \emph{c.f.}~Figure~\ref{v-series}.) and the source
flux is constant (300\,MW).  Smaller panels show: time series of instantaneous
net flux through the MEF perimeter (top); half-hourly current vector (middle
right) with reference arrow indicating true north; and color bar with
temperature anomaly units in $^\circ$C (lower right).  At this moment in the
simulation, despite the substantial anomalies on the north surface, the net
flux has dropped to zero because the flow is near zero.  Animations of the puff
model and Bill Lavelle's diffuse plume model are provided as supplementary
material:
\href{http://econscience.org/scott/pubs/thesis/animations.html}{website}:
econscience.org/scott/pubs/thesis/animations.html} \label{puff-model}
\end{figure}

The time series of $\mathbf{v}$ and $\Delta\theta$ can be used to calculate 4
different types of net horizontal heat flux.  (1)~The $\Delta\theta$ time
series can be multiplied by the corresponding current velocities to obtain a
series of \emph{instantaneous} net flux across the control surfaces.  The
instantaneous flux series can have extraordinarily high variance and include
magnitudes surprisingly different from the steady source flux
(Figure~\ref{puff-model}, top panel}).  (2)~The temperature series can be
averaged before multiplying by the velocity series, or (3)~visa versa; this
results in \emph{temperature-averaged} or \emph{velocity-averaged} flux series,
respectively, both of which have variances and means that are different from
the instantaneous flux series.  Finally, the average of the instantaneous,
temperature-averaged, or velocity-averaged series can be taken, yielding in all
3 cases the same (4)~mean net heat flux, which must equal the magnitude of the
source flux.

Analysis of these modeled flux series reveals a dependence of the net flux on
the mean velocity.  Figure~\ref{flux-comp} shows how the mean and standard
deviation ($\sigma$) of the different net heat flux series (based on a
$\sim$360\,hr simulation) evolve as the mean flow added to the oscillations is
increased in magnitude in the puff model.  The mean of the instantaneous flux
series ($<$T v$>$) is 100\% of the source flux ($H_s$) at mean flows $>$1~cm/s.
At 0~cm/s mean flow, $<$T v$>$ must also be zero because the series is a
product of a the (zero-mean) velocity series.  At $\leqq$1~cm/s the
$\Delta\theta$ gradients are so steep that $\sim$5\% of the heat diffuses
through the control surfaces and is not accounted for.  The averaged flux
series ($<$T$>$v and T$<$v$>$) have mean magnitudes ($<$T$><$v$>$) of
$\sim$75\% of the source flux at mean flows of 1--5~cm/s.  This underestimation
is due to averaging one of the time series prior to multiplication, and
motivates the adjustment in the next section of observed $H_h$ by a factor of
$\sim1.3(=1/0.75)$.  
% Check this via the models:
%The mean values of all flux series approach 100\% $H_s$ at mean flow
%$>$6\,cm/s because the simulated plume is approaching a steady, streaming
%state; at lower mean flows, the oscillatory flow has a stronger influence on
%$\Delta\theta$, generating $\Delta\theta$ series with mean values that are low
%biased.  
The $\sigma$ of the instantaneous flux series is greater than the $\sigma$ of
either averaged flux at all flows.  In the 1--5~cm/s range of mean flow, the
instantaneous $\sigma$ is $\sim$100--200\% of $H_s$.  The $\sigma$ of the
velocity- and temperature-averaged series converge from low-flow values of
150\% and 30\% of $H_s$, respectively, to $\sim$60\% of $H_s$ above 2~cm/s.
Over a range of flow typical of the MEF environment (1--5\,cm/s plus
characteristic oscillation), the model leads to the general expectation that
flux observations near the MEF will yield averaged flux series with $\sigma$
equal to 50--150\% of $H_s$.

\begin{figure}[htbp] \begin{center}
\includegraphics[width=0.8\textwidth]{figs/puff/mof_fluxcomp2.eps2}
\end{center} \caption[Modeled flux statistics over a range of mean flows]
{Means and standard deviations (std) of net horizontal flux ($H_h$) series from
puff model simulations in which mean flow of 0--7\,cm/s were added to
characteristic oscillations.  Statistics are expressed as percentage of source
heat flux ($H_{s}$).  Lower panel shows standard deviations of the 3 types of
net heat flux \emph{series}: instantaneous (T v), velocity-averaged (T$<$v$>$),
and temperature-averaged ($<$T$>$v).  Upper panel shows the mean of the
instantaneous series ($<$Tv$>$) and either averaged series ($<$T$><$v$>$) for
each mean flow.  } \label{flux-comp} \end{figure}


\subsection{Model implications for observed horizontal flux} 

Since our estimates of observed $H_h$ are essentially velocity- and
temperature-averaged fluxes made in an environment where mean flow is
1--5~cm/s, the model indicates that the estimates should be adjusted upward by
a factor of $\sim$1.3~($=1/0.75$).  This adjustment for the uncertainty in a
particular averaging procedure increases the estimate of observed $H_h$ from
10--50\,MW to $\sim$15--65\,MW.  Within the same mean flow range, the standard
deviation of the modeled averaged flux is $\lesssim$100\% of the source flux.
Taking the high end of our $H_h$ magnitudes (65\,MW) as the source flux that
warms the north control surface, the model results suggest that if the Flow Mow
methodology were redesigned to monitor both north and south control surfaces
simultaneously, then the time series of observed averaged horizontal heat flux
would have a maximum standard deviation of $\sim$150\% of 65\,MW, or $\sim$100\,MW.

%Instantaneous approach:

%Address this question: Why can't NoSoMEF T fields from similar phases of tide
%be used to estimate instantaneous flux? 

%Compare puff model Dth to mean and variance Dth observations?

%Discuss and quantify error propagation in HF calculations?  Especially relative to
%the precision of the CTD and $\Delta\theta$...


%\section{Near-bottom spatial and temporal variability}

%\subsection{Puff model analysis}

%Can an analytic approach be taken (Parsons intuition)?

%Consider adding Laurent, JPO 2001 as reference justifying using a $k_h$
%greater than the interior ocean value of Okubo71

%\begin{itemize} \item{WASP} \item{Deepest Current Meter} \item{Comparison with
%Wetzler et al Results} \item{Expected Variance (and Time Series)}
%\item{Weighted Instantaneous Observations} \end{itemize}

%\subsubsection{Temporal variance at No/So/MEF}

%$\Delta\theta$ variance clearly is highest near 2000~m and decreases rapidly
%below, until more uniform variance is encountered in bins below 2100~m.  One
%interpretation is that heat diffuses downward from the strong and variable
%upper plumes to generate corresponding thermal variability in the lower layer.
%Alternatively, the lower layer $\Delta\theta$ variance is generated by
%time-varying currents in the axial valley redistributing heat from steadily
%venting, low $B$ hydrothermal sources.

%These 2 possibilities can be distinguished through a close examination of how
%$\Delta\theta$ means and variance vary, both vertically in any particular area
%and laterally, with increasing proximity to low $B$ sources on the sea floor.
%In the areas north and south of MEF, in each 25~m bin below 2025~m,
%$\overline{\Delta\theta}$ decreases toward the bottom.

%Verify with histograms of near-MEF vs No/So/MEF, including $\sigma$ vals in the
%histogram figures...  Might be worth including histograms from Quebec, too.
%Note that while near-bottom $\overline{\Delta\theta}$ is slightly greater in
%the north than in the south, in almost all of the upper depth bins the opposite
%is true.  

% Also seems like the right place to revisit the clarity of water very near the bottom...

%\subsubsection{Spatial variance from Dive51}

%[Describe spatial patchiness (20~m and 10~min), or the patchiness in Theta-S space?
%FM tops vs HTs vs $S-\theta$ scatter plots?]

%\subsection{Time Series from nearby moorings}

%WASP 2161, 2000, has S; 1995 data?
%analyze cm (Texp,U) time series 

%The near-bottom current meters deployed in 1995 and 2000 were equipped with
%high-precision thermisters.  These sensors provide long and continuous records
%of water temperature variability within the axial valley near the MEF.

%Table summarizing expanded temperature data from current meter moorings

%A) 1000 m N of MEF, WASP mooring Jun?-Sep? 2000
%B) 200 m NE of MEF, RCM5 mooring Jun?-July? 1995
%C) 1200 m S of MEF, RCM5 mooring Jun-Sep? 2000

%Elevation       Depth (m)       Mean(Te)        max(Te) min(Te) std(Te)
%A) 15 mab       2168
%B) 25 mab       2175
%C) 50 mab       2168

%The observed variance in expanded temperature could be due to
%1) Vertical displacement of isotherms by non-hydrothermal forcing
%2) Advection of a temperature field that is heterogeneous due to
%non-uniform (in X or t) input of hydrothermal heat
%3) Advection of a temperature field that is homogeneous at the depth
%of the thermisters, perhaps due to mechanical, near-bottom mixing


%\section{Observed vertical heat flux}

% Begin by saying that the horizontal oscillatory flow surely has some effect
% on the $T$ and $w$ fields on the top surface...  This is a good segue to
% showing variability on top surface...

% It would be nice to have some actual observations in this section (courtesy
% of F. Stahr?)  If this doesn't happen, then move "Correcting" section into
% discussion.

% Dealing with the implementation of the top surface integral
%\begin{itemize}
%\item{Integrated $w_i\Delta\theta_i$}
%\item{Interpolated $w$ and $\Delta\theta$}
%\end{itemize}
%

% Might want to put Correcting for entrainment subsection back here if you include 
% examples of top data and vert heat flux calcs here...


\section{Discussion}

\subsection{Sources of variability in MEF hydrography}

We used both ABE and CTD to monitor the hydrography near the MEF, and
endeavored to survey all sides of the lower control volume at multiple phases
of the tide.  The observed spatial heterogeneity of the plume distribution was
higher than expected and increases with proximity to the MEF.  Temporal
variability on the north and south control surfaces was also surprisingly high.
Although the north surface had greater $\overline{\Delta\theta}$ than the south
surface on average, as expected in the northward mean flow, there were multiple
occasions when the south $\overline{\Delta\theta}$ was greater than at the north
surface.

While the puff model produces highly variable $\Delta\theta$ distributions
through advection and diffusion, it does not include other possible sources of
plume variability that are likely at work within the MEF.  Source flux may
vary, as temperature variability has been observed at some MEF vents on time
scales as short as tidal periods (R.~McDuff, pers.\ comm.), but such variations
are unlikely to alter source fluxes by more than $\sim$10\%.  There are
multiple constant-flux sources with distinct effluent characteristics (as
opposed to the single model source).  Plume rise heights are not uniform (as in
the model), but vary because of distinct source $B$ and changing cross flow
magnitude.  Although I have documented significant coherence in the axial
currents, small amounts of vertical or horizontal shear in the velocity field
will affect the plume distribution and its variability, and would also
influence the calculation of net heat flux.  A final source of variability is
the proximity of other heat sources, beyond the perimeter of the MEF.  Not only
is the diffusely venting Quebec area located nearby, but a major vent field,
Mothra, lies upstream of the MEF.  The puff model indicates that plumes from
either source are likely to be advected intermittently through the MEF.

The puff model does, however, explain major aspects of the observed
hydrographic patterns and variability.  
% This is really the place for presentation of the bow tie plot to SHOW THAT
% THE PUFF MODEL REALLY DOES EXPLAIN SOME OF THE N-S VARIABILITY, and therefore
% is also a good predictor of the statistical properties of measurements made
% on the control surfaces.
%
% It could help to show the one well-correlated model/observ T t-series from
% FM-N15.  This also might be a good place to include the cross-correlation of
% $\Delta_S\theta$ (or just T?) and V at FM-N15.  
%
% The puff in this incarnation can't resolve the small scale heterogeneities,
% however, so that's why we need to talk realistically about expected rise
% heights...
Model results show that tidal pooling and streaming cause dramatic plume
non-uniformity and net heat flux variability (Figure~\ref{puff-model}), even
when the plumes are advected in idealized, exclusively along-axis oscillating
and mean flow.  Modeled plume heterogeneity and variability are even higher in
2-axis currents.  Diffusion, likely enhanced in bottom boundary layers within
the axial valley \cite{thomson+89}, reduces plume heterogeneities over time.
Acting in concert, advection and diffusion distribute thermal anomalies that
are reduced in intensity and more uniform away from hydrothermal sources.
%Even midway between known vent fields, plume induced thermal variability
%rivals that expected from the vertical displacement of isotherms by internal
%waves (cite?). 
On the periphery of hydrothermal fields observed and modeled $\Delta\theta$
magnitude increases.  

% How to prove that intensity and scale of plume features increases with
% distance from known sources?  

Overall, advection and diffusion generate a $\Delta\theta$ distribution that
complicates the task of measuring the net flux from a vent field.  If mixing
rates were greater or there were no mean flow, then the valley would eventually
acquire a uniform hydrothermal anomaly that could be more easily inventoried.
The observed complexity of the hydrography in the valley suggests that the most
successful efforts to measure field-scale fluxes will be those that correctly
establish and simultaneously monitor up- and downstream positions over many
tidal cycles, or otherwise integrate the variability induced by the tides, 
and then use a sophisticated model to help interpret the results.

%ginster+94 estimate $H_s=$122$\pm$61\,MW for the ``Tube Worm'' field, located
%about 2\,km north of the MEF (probably Clam Bed?).

%Despite these potential complications and the observed hydrographic
%variability, the Flow Mow strategy of monitoring the surfaces of a control
%volume led to observed average plume distributions near MEF.  
The high spatial and temporal heterogeneity in the observations are not
resolved by the puff model with a single source, but are both likely caused by
\emph{multiple and distinct} MEF sources venting into variable cross flow.  The
distribution of hydrothermal sources within the MEF is certainly more
complicated than a single central source (Figure~\ref{mefmap}).  Studies of
fluid properties at focused vents evidence higher $T$ and lower $S$ at southern
vents compared to northern vents \cite{butterfield+94}.  During the summer of
2000 all MEF sources had negative salinity anomalies ($\Delta S$): -9 to
-14\,psu for high $B$ sources in the south part of MEF; -4 to -8\,psu in
northern high $B$ vents; and about -0.5\,psu for low $B$ vents in general
(D.~Butterfield, pers.\ comm.).

Numerical models of diffuse plume rise (J.W.~Lavelle, pers.\ comm., see caption
of Figure~\ref{puff-model}) indicate that in typical stratifications, vents
with negative $\Delta S$, no matter the magnitude of their positive source
$\Delta\theta$, will produce plumes that separate from the sea floor downstream
of the source.  When the source properties (measured $S$ and $T$, and estimated
$w$ and $A$) are used to quantify a range of $B$ for MEF low $T$ vents, and the
results are used to initialize the plume rise models (along with a reasonable
range of environmental conditions, namely the buoyancy frequency, N, and
cross flow velocity, U), the range of expected rise heights and plume
thicknesses creates significant temperature anomalies at elevations of 15--35~m
for $\Delta S=+0.5$\,psu and 20--90~m for $\Delta S=-0.5$\,psu.  In cross
flow of 5~cm/s the modeled plumes equilibrate $\lesssim$250~m downstream of
their source.  While plumes from some low $B$ vents have been observed to hug the
sea floor \cite{trivett+williams94,rona+97}, the plumes from slightly fresh MEF
diffuse sources are expected to separate from the sea floor and be nearing
equilibration $>$15\,mas by the time they cross the Flow Mow control surfaces.
These plumes may even rise through the top surface of the control volume (at
100\,mas) during periods of low cross flow, if they are not first entrained by
high $B$ plumes.  

%If currents of $\sim$5~cm/s orthogonal to the sides are assumed to generate
%variability on time scales of $\sim$1000~s (the time it took ABE or the CTD to
%complete a lap horizontally or vertically), then the characteristic dimension
%of thermal patches would be about 50~m, an estimate that is in rough agreement
%with the smallest scale of variation observed on any individual pass.  This is
%also the approximate scale of the smallest features generated by a numerical
%model of a typical MEF diffuse plume (J.W.  Lavelle, pers.\ comm.).

The behavior of focused hydrothermal plumes in uniform cross flow has been
modeled both analytically \cite{middleton+thomson86} and numerically
\cite{lavelle97}.  With some approximation of the typical in-valley
stratification, plumes from MEF focused sources (with the observed range of
source $S$ and $T$) are expected to rise anywhere from 200~m in nearly
quiescent conditions (1~cm/s cross flow) to 100~m in typical peak MEF cross
flows of 10~cm/s ($\sim$5\,cm/s mean flow combined with $\sim$5\,cm/s
oscillatory flow).   Although measurements of 20~cm/s peak cross flows are rare
in the 1995 and 2000 records ($\sim$1 event in 20~days), focused plume rise
during such flow could be reduced to $\sim$50~m.  

These predictions suggest that plumes from both diffuse and focused MEF sources
may be bent over and dispersed in complex patterns at multiple depths by cross
flow, but are unlikely to equilibrate outside of the depths monitored during
Flow Mow.  Taken together, the lower and upper Flow Mow control volumes
(Figure~\ref{fmcv}) span from $\sim$10--400\,mab, or $\sim$2190--1800\,m
depth.  The only unmonitored area of the lower control surfaces was within
5--10~mab, a gap that was avoided to prevent equipment collision with the sea
floor.  With respect to measuring heat flux, if plumes and currents were
uniform from 0--100\,mab, the error introduced from not monitoring the gap
would be 5--10\%.  Given MEF source fluids properties
(\emph{c.f.}~Table~\ref{B_table}), however, modeled plume rise heights suggest
that the heat flux through this gap is negligible compared with the flux
through the control surfaces in the 5--100~mab depth range.  

The small-scale heterogeneities observed at different depths on the control
surfaces probably originate when diffuse sources with different $B$ and depth
(10--15~m vertical separation) vent into a mean cross flow, generating plumes
with distinct rise heights.  When tidal oscillations also influence the
dynamics and distribution of plumes, high temporal variability becomes the
norm, especially when observations are made close to the hydrothermal sources.
During peak cross flows, even plumes from high $B$ sources may bend over far
enough to transit the upper $1/2$ of the side surfaces of the lower control
volume.


\subsection{Corrections to horizontal and vertical heat flux estimates}


The heterogeneous hydrography observed within the axial valley implies that the
fluid entrained by rising MEF plumes may have variable temperature.  This
complicates the interpretation of heat flux measurements made on the top
surface of the Flow Mow control volume because relating measured anomalies to
source heat flux through 1-dimensional plume theory requires an assumption of a
unchanging background $\theta$--$S$ relationship in the entrained fluid.  

However, the consistent northward flow within the axial valley means that fluid
entrained by MEF plumes enters the field predominantly through the south side
of the control volume.  The mean $\Delta\theta$ of the fluid as it enters the
lower Flow Mow control volume within 100\,mab is 
%at the SoMEF station and on the south control surface 
${\overline{\Delta\theta}}_S\simeq0.05^\circ$C.  As with the horizontal
isohaline heat flux $H_h$ in Equation~\ref{hh}, the vertical isohaline heat
flux $H_v$ can be calculated by subtracting ${\overline{\Delta\theta}}_S$ from 
%each of $n$ observations of 
the mean $\Delta\theta$ measured on the top surface
(${\overline{\Delta\theta}}_v$) by ABE: \begin{equation}
%H_{v} = A_v \rho c_p \sum_{i=1}^{n}{ w_i (\Delta\theta_i -
%{\overline{\Delta\theta}}_S) },
H_{v} = \rho c_p A_v w ({\overline{\Delta\theta}}_v -
{\overline{\Delta\theta}}_S) \label{hv} \end{equation} where $w$ is vertical
velocity and $A_v$ is the area of the top surface.  Equation~\ref{hv} yields
$H_v=550\pm$100\,MW for observed $w$ and ${\overline{\Delta\theta}}_v$
\cite{stahr+03} and involves the assumption that fluid entrained at any depth
within $\sim$100~mas has the same average $T$ and $S$.  This assumption is
justified partially by the observation of intermittent well-mixed layers
$<50$\,mab.  and partially by the observation during horizontal CTD tows
through the MEF at depths of 30--60~mas that the $\Delta\theta$ between buoyant
plumes is consistently $0.05\pm0.02^\circ$C
(\emph{c.f.}~Section~\ref{sec:hts}).  Altering the value of
${\overline{\Delta\theta}_S}$ by 0.01$^\circ{C}$ changes the vertical flux
estimate by $\sim$10 MW.

Prior to comparing Flow Mow heat flux estimates with historic estimates (in the
next section, and in Chapter~\ref{ch:conc}), the isohaline heat fluxes $H_h$
and $H_v$ (calculated with $\Delta_S\theta$) must be converted to level heat
fluxes $H_h^*$ and $H_v^*$ (calculated with $\Delta_z\theta$) in accordance
with Equation~\ref{hhs}.  Multiplication of $H_h=$65\,MW and $H_v$=550\,MW by
the correction factor $C=1.17$ yields $H_h^*=76\pm$114\,MW and
$H_v^*=643\pm$116\,MW, respectively.


\subsection{Partitioning of power}
\label{sec:pop}

Best estimates of $H_h^*$, $H_v^*$, and $H_f$ can be used to infer how MEF heat
flux is partitioned between focused and diffuse vents.  This is accomplished by
assuming that the net heat flux through the sea floor ($H_f+H_d$) into the
lower control volume (\emph{c.f.}~Figure~\ref{fmcv}) is equal in magnitude to
the net heat flux out ($H_v^*+H_h^*$).  The assumption leads to an expression
for the steady state heat budget in the lower Flow Mow control volume
(Figure~\ref{fmcv}) in which all heat fluxes are based on level-to-level
potential temperature anomalies ($\Delta_z\theta$): \begin{equation} H_d + H_f
= H_v^* + H_h^*.  \label{hcons} \end{equation}

To calculate $H_d$, the heat flux from diffuse sources, I take
$H_f=615\pm$123\,MW, the best estimate of \citeN{ginster+94}
(\emph{c.f.}~Section~\ref{sec:hist}).  This value of $H_f$ has the advantage of
a relatively low, well-determined uncertainty (20\%) and is near the middle of
the range of $H_f$ (359--1224\,MW) estimated by \citeN{bemis+93}.  
A limitation of this approach is that $H_v^*$ and $H_h^*$ are based on
measurements made in 2000, while $H_f$ is based on measurements made in 1988.
Since more recent independent estimates of $H_f$ are not available for the MEF,
I proceed with these values.  Solving Equation~\ref{hcons} for $H_d$ and
substituting $H_f=615\pm$123\,MW along with the Flow Mow estimates of
$H_v^*=643\pm116$\,MW and $H_h^*=76\pm114$\,MW yields $H_d=$104$\pm$253\,MW.
This magnitude of $H_d$ is close to the preliminary results of
\citeN{johnson+02}, in which point measurements of vertical heat flux in
diffuse vents were extrapolated to the total area of diffuse flow within the
MEF, mapped by acoustic scintillation tomography, generating an estimate of
$H_d\sim$150\,MW. 

Thus, MEF heat flux is inferred to be partitioned between diffuse and focused
sources in a ratio of 104:615, or about $H_d$:$H_f=$1:6$=$0.17.  This
partitioning of power contrasts with the idea that diffuse flux generally
dominates focused flux in a ratio of about 10:1 in the MEF \cite{schultz+92},
10:1 in the ASHES vent field at Axial volcano \cite{rona+trivett92}, or 2:1 in
the north Cleft vent field on the Juan de Fuca ridge \cite{baker+93}.  Given
the combined standard uncertainty of the inferred value of $H_d$ (50\%) and the
uncertainty in the best estimate of $H_f$ (20\%), the ratio $H_d/H_f$ could be
as high as 0.72 ($=(104+253)/(615-123)$).  Such a value would be comparable the
ratios of about 1:1 inferred by \citeN{lavelle+01} at Axial sea mount further
south on the Juan de Fuca ridge.  

Assuming that only diffuse sources are responsible for the observed horizontal
flux $H_h^*=$76\,MW, then an additional implication of $H_d$ having a magnitude
104\,MW is that roughly $1/4$ of $H_d$ is entrained into rising plumes within
the MEF perimeter.  The other $\sim3/4$ must escape the field laterally.  


