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\title{Controls on Exchange through a Tidal Mixing Hotspot at an
  Estuary Constriction}



\author[1]{Susan E. Allen}%
\author[1]{Nancy K. Soontiens}%
\author[1]{Michael Dunphy}%
\author[1]{Elise M. Olson}%
\author[1]{Douglas J. Latornell}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 27, 2024}


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\subsection*{A version of This Work has been submitted to\\ {\it J. Physical Oceanography}}
 This Work has not yet been peer-reviewed and is provided by the contributing Author(s)
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   the copyright owner. Copyright in this Work may be transferred without further notice.
   
\section*{Abstract}

Deep estuaries are often separated from the open ocean by sills and
constrictions. These constrictions are areas of intense mixing often
dominating the total estuarine mixing. The amount and depth of the
estuarine exchange depends sensitively on the mixing and the densities
of the waters on the two sides of the mixing region. Thus, the
density, nutrient concentration, oxygen saturation, and dissolved
inorganic carbon content of the incoming estuarine flow depend on
local tidal mixing processes and large scale buoyancy dynamics. We
have investigated this process using a numerical model (SalishSeaCast)
of the Salish Sea on the West Coast of North America, straddling the
Canada/USA border. The region receives considerable freshwater
dominated by the outflow of the Fraser River. The Fraser River first
flows into the deep Strait of Georgia but the freshwater must traverse
the strongly tidally mixed shallower passages through the Gulf/San Juan Islands
before it reaches the Pacific Ocean. The model correctly reproduces
the deep water flow into the Strait of Georgia as evaluated against
Ocean Networks Canada (ONC) four bottom-mounted, continuously
recording, conductivity-temperature instruments which capture this
incoming flow. Using a four-year hindcast from the model we determine
the amount, depth and position of the outflow and inflow. We show that
95\% of the variance of the 4-day average baroclinic flux through the
tidal mixing region can be explained by the density difference across
the region and a Richardson Number based on the tidal velocities.  The
outgoing flux includes both surface and intermediate waters and the
incoming flux includes both intermediate and deep waters.  Laterally,
fluxes into and out of the Strait of Georgia and across Victoria Sill
show the impact of the Coriolis force and local bathymetry.   
\section*{Introduction}

A classic lock exchange experiment in a laboratory separates two
different densities by a lock that is removed ({\it e.g.} (missing citation)).  As the lock is
removed, the lighter water flows over the heavier and the heavier
water flows under the lighter.  These two gravity current flows travel
quickly, on the order of the internal wave speed, quickly
redistributing the density.  Performing the experiment on a rotating
table reduces the lateral width of the gravity currents but does not
significantly change their speed (missing citation).  On the other hand, introducing
turbulence, say in the form of a bubble region, breaks up the gravity
currents and significantly increases the time for density exchange (missing citation).

These basic dynamics, a contrast in densities driving exchange, and
turbulence reducing the rate of exchange, are expected to explain real
oceanographic situations.  Here we look at their application to an estuarine system with a highly constricted, very
turbulent tidal mixing ``hotspot''.  Using the results from a well-resolved
three-dimensional ocean model, we ask if the laboratory dynamics apply
and what they tell us about the exchange flow for this estuary.

In the oceanographic literature, the estuarine exchange problem has a
long, and mostly separate history from these laboratory studies.  In a
real estuary, the problem is greatly complicated by the presence of
tides.  We cannot remove them (even from our model) as they are
determining the mixing in the system.  However, at any given time, the
instantaneous velocity is largely determined by the tides, advecting water in
and out.  A traditional way to determine the net estuarine transport
from observations
is to assume two layers and to use Knudsen's Relations, that is, conservation of water and
salt. However, adding symmetrical mixing, as opposed to just
entrainment into the upper layer, makes the system under-determined.
One can use temperature and heating versus salinity to remove this
ambiguity and then invert temperature and salinity profiles to
determine the transport assuming a layer structure (missing citation).

The water advected in and the water advected out by the tides may be largely the same, and
what we want to extract is the difference. From model results, one can
analyze time averages, giving the tidally averaged
velocities and tidally averaged salinity.  However, this process neglects the
strong correlations in the fluctuations. A more complete way to do this is to
bin the water and salt fluxes by salinity bin and calculate TEF or
total exchange flow (missing citation); (missing citation).  This
method captures both the estuarine exchange flow and the transport due
to the tides, and thus is a total or maximal exchange
(missing citation).

The estuary we will consider is the large, semi-enclosed Strait of
Georgia (SoG), which is connected to the Pacific Ocean through a western and
a northern entrance (Figure~\ref{fig:map}).  The primary flow is
through the western entrance; the northern entrance is small and flux
and exchange
there are significantly smaller than at the west(missing citation); (missing citation); (missing citation); (missing citation).  The major source of fresh water is
the Fraser River, about 60\% of the total to the SoG (missing citation) which enters the
SoG near its south end.  At the south end of the SoG are the Gulf/San
Juan Islands that form lateral constrictions.  The water through this
region is also shallower (Figure~\ref{fig:transects}).  Thus, tidal
flows are high, up to
4.5 m s$^{-1}$, and turbulent mixing is strong (Figure~\ref{fig:transects}).  Water exiting the SoG
flows through this region and then into the relatively straight Juan
de Fuca Strait (Figure~\ref{fig:map}), although there is a significant
sill, Victoria Sill, at the eastern end of Juan de Fuca Strait (Figure~\ref{fig:transects}).

Exchange through this region has been studied through observations (missing citation); (missing citation); (missing citation); (missing citation)
and models (missing citation); (missing citation).  Exchange is
seasonally variable with pulses of freshwater exiting Juan de Fuca
Strait during the Fraser River Freshet, weak neap tides, and winds to
the south in SoG (missing citation).  Deep water renewals similarly occur
during neap tides during spring through fall
(missing citation); (missing citation); (missing citation).  
The observed dense water cascading from Boundary Pass Sill into
the deep SoG has been successfully
explained as a gravity current (missing citation); (missing citation).

Net fluxes 
have been estimated at 46 mSv out from Boundary Pass or 114~mSv in
from Victoria Sill (missing citation).  These
estimates are based on mass, salt and heat balances and are
necessarily coarse in their vertical resolution and do not include
lateral resolution. Using the TEF method the flux
into the SoG is estimated as 83 mSv
(missing citation), updated to 70 mSv (missing citation).  These two estimates include both the
estuarine component and the net tidal impact.  Estimates not
including the tidal impact are lower (28 mSv, (missing citation)).

However, all these estimates include both the flux that transitions
the turbulent region and flux that is recycled within it. In particular,
the flux into and out of Haro Strait is different in (missing citation)
because much of the deep flow into Haro Strait is entrained into the
surface flow and exits back out in the surface outflow to Juan de Fuca
Strait. Indeed, although the exchange is maximum during neap
tides, the maximum residual flow actually occurs at spring tides due
to this entrained flux (missing citation). This short-circuited flow is referred to as the reflux ({\it
  e.g., } \citeauthor{MacCreadyetal2021}, \citeyear{MacCreadyetal2021}).  Here we will use a Lagrangian tracking method that allows us
to separate the reflux and focus on the flux that travels through the
mixing region. This method does include the effect of the tides.

Typically the SoG is divided into three layers: a
surface layer above 50~m, an intermediate layer, and a deep layer below
200~m (e.g. \citeauthor{Masson2002}, \citeyear{Masson2002};
\citeauthor{Pawlowiczetal2007}, \citeyear{Pawlowiczetal2007}).  Looking at the
monthly climatology in the central SoG these depths would correspond to salinities of 30
g kg$^{-1}$ (range 29.9 g kg$^{-1}$ (July) to 30.4  g kg$^{-1}$ 
(November)), for 50~m and 31.2 g kg$^{-1}$ (range 31.0 g kg$^{-1}$ (May) to 31.4  g
kg$^{-1}$ (October)) for 200~m.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/map-for-estuarine-exchange/map-for-estuarine-exchange}
\caption{{\label{fig:map} Map of the region with the model domain shown by the grey
  box.  The two open boundaries are the western boundary at the mouth
  of Juan de Fuca Strait and the northern boundary just north of where
  the bottom deepens in Johnstone Strait.  The red line shows the
  thalweg, the deepest line through the model.  Colour is the surface
  salinity averaged over one hour.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/allen-thalweg-19aug19/allen-thalweg-19aug19}
\caption{{\label{fig:transects} \small Upper panel: Salinity along the thalweg shown in
  \protect{Figure~\ref{fig:map}} from west to north averaged over one hour at same time
  as the surface plot. Marked is the beginning and end of the mixing
  region through the Gulf/San Juan Islands.  Note the similar mixing
  region to the north end of the domain through Discovery and
  Johnstone Straits.  The red star shows the location of Boundary Pass
  where the tidal amplitude was extracted. Lower panel: Detail of
  \protect{Figure~\ref{fig:map}} for the Gulf/San Juan Islands region
  showing the location of the transects used for the Ariane analysis.
  The transects align with the grid:  cyan: Gulf Islands, green: Point
  Roberts, pink: Victoria Sill, yellow: Puget Sound, white: Deception
  Pass. White symbols are VENUS observation nodes: circle: Central,
  square: East, triangle: BBL, star: DDL. Red Star: Boundary
  Pass where tidal amplitude was extracted. Red square and red circle:
  point on Point Roberts Line
  and point on Victoria Sill Line, respectively, between which the density
  forcing was calculated. Contours show salinity at 77 m
  averaged over one hour with darkest green showing water less than
  77~m deep.%
}}
\end{center}
\end{figure}

The exchange of salt and heat between the estuary and the open ocean
is a classic problem as discussed above.  Understanding the fluxes of
nutrients (missing citation); (missing citation) and carbon (missing citation) in this estuary has clear
applications to the potential for eutrophication and for the changing pH
climate of the estuary.  The
strong nutricline and carbon-cline, not directly aligned with the
halocline in the estuary, imply that the geometry
of the exchange flow will significantly impact the flux of these
tracers.  Thus, we will quantify the lateral and vertical geometry of
the flux as well as its total value.

In the next section we briefly review the theory of density exchange
through a turbulent region.  Then we describe our methods including
the hydrodynamic model, SalishSeaCast, and the Lagrangian analysis method.
Our results include a model evaluation, a comparison of our results to
the theoretical exchange flux, its properties and geometry. We
discuss our results in comparison to previous Salish Sea studies and
describe the potential impacts of the geometry of the flux on the
transport of biogeochemical tracers.  Using the theoretical model, we
discuss the robustness of the exchange flow to climate change.  We
finish with our
summary and conclusions.

\section*{Theory}

A two-layer exchange flow between water of density $\rho_1$ and
$\rho_2$ through a depth $H$ is given by:

\begin{equation}
u = \frac {1}{2} (g' H)^{1/2}

\end{equation}
where $g' = g (\rho_2 - \rho_1)/\rho_o$, $g$ is the acceleration due
to gravity, and $\rho_o$ is a mean/typical density
(missing citation).  Thus the baroclinic exchange flux should
be

\begin{equation}
U =  (g' H)^{1/2} H W

\end{equation}
where $W$ is the width of the channel.

(missing citation) calculate a Richardson number, $Ri$, to characterize
the mixing in their system and show that the square-root of it gives
the tracer flux through the system.  They calculate the $Ri$ based on
the density gradient and the velocity fluctuations.

Thus, we propose that the baroclinic flux should a function of the
density forcing (related to $(g' H)^{1/2}$) and inversely to the
mixing characterized by $Ri^{1/2}$.  However, in the Salish Sea we don't have
two layers and instead we characterized the density forcing as the
difference in hydrostatic pressure between the SoG and Victoria Sill.
For the velocity fluctuations we use the tidal flow measured in
Boundary Pass (Figure~\ref{fig:transects}).  Thus our density forcing is:

\begin{equation}
D_F^2 =\left. \frac{g}{\rho_o} \int_{zb}^0 \rho \, dz \right|_{PR}^{VS}

\end{equation}
where $zb$ is an appropriate depth, taken as the depth at Victoria
Sill as it is shallower, VS is the point on
Victoria Sill and PR is the point on the Point Roberts transect. Our
Richardson number
  is:

\begin{equation}
Ri = \frac{D_F^2}{U_T^2}

\end{equation}
where $U_T$
is the velocity at 10~m depth at Boundary Pass
(Figures~\ref{fig:transects}).

From the theoretical considerations we would expect the baroclinic
flux through Boundary Pass would be:

\begin{equation}
{\cal F} \propto D_F Ri^{1/2} H W
\label{eq:theory} 

\end{equation}

\section*{Methods}

Tides are strong in Haro Strait and many other estuaries.  As one investigates
exchange, there are a number of methods for removing the
back-and-forth movement of the tides as discussed above.  Here we
will use a Lagrangian perspective: tracking parcels from one end of
the mixing constriction to the other.  Thus, using the results from a
three-dimensional circulation model for velocities, we use a
quantitative Lagrangian method to track fluxes from the estuary to the
ocean and track fluxes from the ocean to the estuary.  This method
gives us water flux, salt flux, heat flux, time of passage and the
geometry of the inflows and outflows through our chosen start and end cross-sections.

\subsection*{SalishSeaCast}

SalishSeaCast is a configuration of the Nucleus for European Modelling
of the Ocean (NEMO) community model (missing citation).  The model domain includes the
SoG, Juan de Fuca Strait, Puget Sound and the southern
part of Johnstone Strait (Figure~\ref{fig:map}).  It is a primitive
equation, baroclinic model with a coupled biological model.  The tides
and storm surge (missing citation), mixing and flow through
Haro Strait (missing citation), and nutrient fluxes
(missing citation) have been previously presented.  This version of
the physical model is the same as presented in (missing citation). % 201806 and 201812 have same physics, different biological parametrization.  In particular in 201806 the nitrate drifted.

The model has a regular quasi-rectangular grid turned 29$^o$
counter-clockwise (Figure~\ref{fig:map}) with the long-axis from
Southeast to Northwest; we will call this along-strait.  The
along-strait grid spacing is about 500~m; the across-strait spacing is
about 440~m.  Vertical grid spacing is 1~m down to about 20~m depth
and then increases with deepest grid cells 27~m thick.  The surface focused
resolution is to allow good simulations of the biological productivity that, due to
strong stratification, is largely above 5~m.

The model uses a split barotropic (2~s) baroclinic (40~s) explicit
scheme.  The advection is Total-Variation-Diminishing (TVD) with a 2~s
time step in the vertical and 40~s in the horizontal.  The horizontal
mixing is based on a constant Laplacian mixing of 1.5 m$^2$~s$^{-1}$. The
vertical mixing uses the k-$\epsilon$ GLS scheme (see
(missing citation)).  Details of differences from
(missing citation) are in the methods section of (missing citation).

The model is forced from the atmosphere, by rivers and through the two
open boundaries.  The winds, incoming solar radiation, incoming long
wave radiation, air temperature and humidity are taken from the
Environment and Climate Change Canada High Resolution Deterministic
Prediction System (HRDPS) (missing citation) and applied to the model using the CORE
scheme (missing citation).  There are over 150 river ``mouths'' in the model.  Other than
the Fraser River, these rivers are grouped into watersheds and forced
according to the (missing citation) climatology.  The Fraser River is
extended in the model (Figure~\ref{fig:map}) to allow stronger tidal
flows at its mouth.  At its head, river flux is a combination of the
river flux measured at Hope (missing citation) and the (missing citation)
climatology for tributaries entering below Hope.

The tides in the form of 8 tidal constituents are forced at the two
open boundaries.  The sea surface heights and velocities were
originally taken from webtide predictions (missing citation) and then tuned to best
match observations at Victoria, Campbell River and Point Atkinson, near
Vancouver (missing citation).
Sea surface height at the western boundary is taken from observations
at Neah Bay with our resolved tides and nonlinear tides removed.  Sea
surface height at the northern boundary is equal to that at the
western with a positive 0.098~m offset based on (missing citation).  Barotropic
velocities are calculated by a Flather scheme to be in balance with
the applied sea surface height. As the estuarine
exchange flow is expected to be set inside the domain, we use Orlanski
radiation conditions with nudging and sponge following (missing citation).  We had to
implement the sponge as it is not standard in NEMO.

A version of the model is run
operationally.  This manuscript is based on a hindcast (v201806).  The model was run coupled to the biological model SMELT
(missing citation) meaning that the light penetration into the water, and thus the solar
heating, is a function of the modelled chlorophyll concentration.
Otherwise, the biological model does not impact the physical model
which will be our focus in this manuscript.  The model was started in
November 2014 with our best estimate of the temperature and salinity
(and thus stratification) at that time from previous model runs and at
rest. It was run through 14 months to ``spin-up'' and then run from
January 2014 - May 2019.

\subsection*{Ariane}

Ariane is a Lagrangian tracking algorithm that interpolates velocities
at each time step (one hour here)
in each model grid cell by three-dimensional linear interpolation and
then tracks particle movements exactly through the cell to the next
cell boundary or the end of the time step (missing citation).  Flows resolved by the
underlying numerical model (here SalishSeaCast) advect the particles.
There is no representation of sub-grid-scale mixing or advection.  In
quantitative mode, Ariane seeds a full set of particles at every time
step over the source transect and calculates the flux represented by
each particle.  These flux representing particles are then tracked to
an exit transect, or the original transect (missing citation); (missing citation).

We chose transects to completely bracket the mixing region and to
align to the grid to reduce interpolation errors.  Considering
salinity along the
thalweg through the model (red line on Figure~\ref{fig:map}), we see
the mixing region as steeply sloping isopycnals between the Victoria
Sill and the entrance into the deep SoG (Figure~\ref{fig:transects}).  Thus we chose
our primary transects as Victoria Sill (the ocean end of the mixing
region) and Point Roberts (the SoG end of the mixing region).  To
close the SoG end, we continue from Galiano Island to Salt Spring
Island and then from Salt Spring Island to Vancouver Island; we call
this the Gulf Island transect.  Then we run a transect across Puget
Sound and add a very short Deception Pass piece: these latter two will
be combined and called Puget Sound.

We run Ariane in quantitative mode for each day in the simulation,
following the particles for a maximum of 30 days.  For each transect,
we run both forward in time and backward in time.  We use
SalishSeaCast results from December 2014 and January 2019 to complete
this full 4 year time series. We record the flux
of water that arrives at each of the other transects, its starting and
finish depth, lateral location, time of passage, temperature and
salinity.

Even after carefully adding variable volume handling code to Ariane we
had three separate days where one parcel on one transect failed.  We patched these three days
by interpolating from the day before and the day after.

\subsection*{Ancillary Variables}

The mixing box models suggest that the transport through the mixing
region should be forced by the density difference between the two ends and
restricted by the mixing strength.  In addition,
(missing citation) show the importance of winds in the SoG
pushing the Fraser River plume north or south.  If pushed south, the
density at the northern end of the mixing region should be reduced.

\paragraph{Density}
As a measure of the density difference we depth integrate the density
from the surface to 160~m depth at a central location on
Victoria Sill and subtract the depth
integrated density at a central point on the Point Roberts transect.  The Victoria Sill line is
shallower, so we only integrate Point Roberts down to the same depth.

\paragraph{Mixing Strength}
As a measure of the tidal mixing strength we use the along-strait speed
squared taken from the model, over the sill in Boundary Pass
(Figure~\ref{fig:transects}).  The speed squared is taken at 10~m
depth from the output hourly averages.  We then low-pass filter the
squared speeds using a 4-day rolling mean.

\paragraph{Winds}
We use the 10-m winds at the grid point closest to Sand Heads
Lighthouse Station from the HRDPS model. Sand Heads were the winds
used by (missing citation). The model winds and measured winds
agree well here (missing citation).  Here we use the rolling
4-day mean wind speed.

\paragraph{Sea Surface Height at Neah Bay}
In order to simulate storm surge (missing citation), we use the
observed sea surface height at Neah Bay
(\url{https://tidesandcurrents.noaa.gov/waterlevels.html?id=9443090}).
This sea surface height directly forces sea surface height in the
model.  In addition, variations in sea surface height at Neah Bay
reflect the along shore winds and upwelling/downwelling on the West
Coast.  Here we use the rolling 4-day mean sea surface height.

\paragraph{Freshwater Flow into the Strait of Georgia}
The freshwater into the model is based on daily observed flows of the
Fraser River at Hope and a climatology for all other rivers, including
the Fraser west of Hope.  For comparison to our results, we extracted
the fresh water into just the SoG (from just south of
Discovery Passage to the southern edge of Boundary Pass). We use the 4-day rolling average.

\section*{Results}

\subsection*{Model Evaluation}

SalishSeaCast model results have been evaluated in several previous
 manuscripts and further evaluations are available on the website (\url{salishsea-meopar-docs.readthedocs.io}).  In
 particular, tides and storm surge (missing citation) and the
 nutrients and phytoplankton (missing citation) are well simulated by
 the model.  Here we focus on evaluation of the temperature and
 salinity field, in general, and of the incoming deep currents into
 the SoG, in particular.

Temperature and salinity are measured about four times per year on the
Institute of Ocean Sciences Strait of Georgia Cruises using a
Conductivity-Temperature-Depth (CTD) instrument
(missing citation). Their data are available from \url{data.cioospacific.ca}.  We compare the values from the observations to
values from the model at the same hour and closest model point
(Figure~\ref{fig:TS}).  The model closely simulates temperature and
salinity with the largest errors at low salinities due to the Fraser
River plume being shifted north in the model.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/EvalDFOTSforSusan/EvalDFOTSforSusan}
\caption{{\label{fig:TS} Temperature (left) and Salinity (right) comparison to
  data. Color shows the count of points (observation matched to
  model). Thin black line is one-to-one.
Temperature bias -0.065$^o$C, RMSE 0.46$^o$C, Willmott Skill
  score 0.965.
Salinity bias 0.13 g~kg$^{-1}$, RMSE 0.49~g~kg$^{-1}$, Willmott Skill
score 0.968.  Agreement is very strong except that the Fraser River
plume is not quite in the correct place and the northern SoG is a little too fresh.%
}}
\end{center}
\end{figure}

Ocean Networks Canada (ONC) has maintained up to 4 bottom mounted
CTD's in the Southern SoG directly in the path of the incoming deep
water from the mixing region (data available from \url{data.oceannetworks.ca}).  We compared temperature and salinity (TS) at
the bottom to these observations over the four years
(Figure~\ref{fig:venusnodes}).  Not only does the model not drift
compared to the observations, it reproduces the interannual, seasonal
and tidal fluctuations in TS.  In particular, it reproduces the strong
deep water renewal pulses at the lowest node, showing incoming cold
water in the early part of the year, switching to warm water in the
late summer to fall.  Almost all pulses are salty.  Higher, at the
East Node, the model reproduces the high temperature fluctuations in
the summer compared to lower variability in the winter.

Like most
models we see that processes such as the pulses of deep water renewal
are represented but have slightly smaller amplitude in the model.
Importantly we see the timing well reproduced. 
Thus we are confident that the model is reproducing the key features
of the mixing and estuarine circulation and is a tool
`fit-for-purpose' to analyze the controls on exchange.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/ONC-comparison/ONC-comparison}
\caption{{\label{fig:venusnodes} Model (pink) to data (green) comparison at VENUS Stations.  Left side panels:
  salinity. Right side panels: temperature.  Comparisons are for VENUS
  Central (300~m depth), East (160~m depth), BBL (140~m depth) and DDL
  (110~m depth) nodes (Figure~{\protect\ref{fig:transects}}).%
}}
\end{center}
\end{figure}

\subsection*{Water Flux}

We calculate the deep flux from Victoria Sill to Point Roberts and the
Gulf Island line, both by tracking forward from Victoria Sill and by
tracking backward from Point Roberts and Gulf Island lines
(Figure~\ref{fig:transports}a, Table~\ref{tab:meanfluxes}).  The mean
fluxes agree (Table~\ref{tab:meanfluxes}) but the time series are not
identical (Figure~\ref{fig:transports}a).  The difference is due to the
timing.  It takes about 2 weeks for parcels to
move from one line to the other.  The forward flux on a given date is
all parcels that left Victoria Sill on that date.  The backward flux
on a given date is all parcels that arrived at Point Roberts or the
Gulf Islands line on that date.\selectlanguage{english}
\begin{table}[bh]
\begin{tabular}{|ccc|cccc|c|}
\hline
{\bf From} & {\bf To} & {\bf Direction} & {\bf 2015} & {\bf
                                                            2016} &
                                                                    {\bf
                                                                    2017}
  & {\bf 2018} & {\bf Mean}
  \\
&&& (mSv) & (mSv) & (mSv) & (mSv) & (mSv) \\
\hline
Victoria& Point Roberts & Forward & 23.3 & 24.0 &24.2& 23.2 & 23.7\\
Victoria& Point Roberts & Backward &23.2& 23.7& 24.1 &  23.6 & 23.6\\
Victoria& Gulf Islands & Forward & 2.4 & 2.1 &2.3 & 2.1 & 2.2\\
Victoria& Gulf Islands & Backward &2.4& 2.1&2.3 & 2.2 & 2.2\\
\hline
Point Roberts & Victoria & Forward & 23.9 & 24.2 &24.0& 23.2 & 23.8\\
Point Roberts & Victoria & Backward & 24.0& 24.0  &24.0& 23.1 & 23.8\\
Gulf Islands & Victoria & Forward & 8.1 & 7.9 & 8.3& 7.9 & 8.0\\
Gulf Islands & Victoria & Backward & 8.0 & 7.8 &8.2& 7.7 & 7.9\\
\hline
\end{tabular}
\caption{{Annual Averaged Fluxes and Four Year Mean}}
%Transport-Analysis
\label{tab:meanfluxes}

\end{table}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/transports/transports}
\caption{{\label{fig:transports} Transports calculated using Ariane.  Panel
    (a) and (b), deep water from Victoria Sill to Point Roberts
    and the Gulf Islands.  Panel (c) and (d), shallow water from Point
  Roberts and the Gulf Islands to Victoria Sill.  Flux is calculated
  both by tracking particles forward (time is when they start) and by
tracking particles backward (time is when they arrived). Panel (b) and
(d) show the time scale shifted to bring the forward and backward time
series together based on the time it took the particles to travel
between the transects (see text).  The red line is the mean flux, averaged between the
two shifted time series.%
}}
\end{center}
\end{figure}

We can correct for the difference in timing (the lag) and calculate a mean timing
value based on the mean time of passage for all parcels in a given
direction on a given day. However, the lag between the two time series
is not constant; it increases with the time of passage.  We estimate
the lag as a linear function of the mean time of passage and chose the
intercept and slope to maximize the lagged correlation. If we then shift the forward time series
forward by half that lag and the backward time series backward by
half that lag
we see strong correlation between the time series
(Figure~\ref{fig:transports}b).
For the
deep flux from Victoria Sill to Point Roberts and the Gulf Island
line, the intercept and slope are -19 days and 1.8, respectively.
Averaging these two time series then gives us mean time series
centered on the mean time a parcel was between the two lines
(Figure~\ref{fig:transports}b).
We can do the same for the shallow flux from the Point Roberts and
Gulf Island lines to Victoria Sill (Figure~\ref{fig:transports}c and
d). The intercept and slope are found to be -8.2 days and 1.3, respectively.

The mean fluxes (Table~\ref{tab:meanfluxes}) over the four years from Victoria Sill (25.9 mSv) and to Victoria
Sill (31.8 mSv) do not match.  There is more flux out of the SoG than
into it across Victoria Sill. Of course, there are a number of other
fluxes across the Point Roberts line that need to be considered: the
flux exchange with Puget Sound, the freshwater flux and the through
flux from Johnstone Strait to JdF.  The transport from Point Roberts
to Puget Sound is 4.2 Sv where as the transport in the other
direction is 2.1 Sv (Table~\ref{tab:sup_PSmeanfluxes}). Thus there is a
net transport of 2.1 mSv from the SoG to Puget Sound.
There is also a net addition of freshwater in the SoG from
the Fraser River  and other rivers of about 5.7 mSv
(missing citation). So net flux to Victoria Sill (5.9 Sv) plus
net flux to Puget Sound (2.1 Sv) is partially balanced by the
freshwater flux (5.7 Sv).  This implies that the 
higher sea level height at the north end of Vancouver
Island generates a net flux from Johnstone Strait to Juan de Fuca
Strait of 2.3~mSv.

The mean fluxes can be compared to the mean annual flux of freshwater (5.7 mSv) with the southward flux being 5.5 x the Fraser River flux
and the northward flux being 4.5 x the Fraser River flux.  These give
an estuarine amplification of only 5 which is very low, emphasizing
that we are only considering parcels that transit the whole mixing
region, not parcels that enter in the deeper layers, are mixed
into the upper layer and exit again without reaching the SoG.

To further investigate the dynamics of the flow, we combine the
shallow flux and deep flux by adding and subtracting to get the
barotropic flux and baroclinic flux (Figure~\ref{fig:ancillary} a \& e
and Figure~S\ref{fig:sup_ancil} a).  The barotropic flow is small, with positive
southward flux due to the sea surface tilt between Johnstone Strait
and the mouth of Juan de Fuca Strait.  The baroclinic flow is the
exchange flow.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Ancillary/Ancillary}
\caption{{\label{fig:ancillary} Two-year (left-panels) and six-month (right panels) time
  series of a \& e: barotropic and baroclinic flux between Point
  Roberts and Victoria Sill from the Ariane analysis, b \& f: four-day
running mean tidal speeds at Boundary Pass and the integrated density
difference between Point Roberts and Victoria Sill, c \& g: the
freshwater flux into the model and d \& h: the four-day running mean
wind velocity near Sand Heads and the four-day running mean sea
surface height at Neah Bay.  Full four year timeseries are in
Supplementary Figure~\protect{\ref{fig:sup_ancil}}.%
}}
\end{center}
\end{figure}

The baroclinic flux is well correlated with the density difference between
Victoria Sill and Point Roberts, particularly at the seasonal time
scale (Figure~\ref{fig:ancillary} a, b and Figure~S\ref{fig:sup_ancil} b).  On a short time scale, we can see a strong negative
correlation between the baroclinic flux and the tide speed squared
(Figure~\ref{fig:ancillary} f).  The river runoff is not as well
correlated to the baroclinic flux as the density forcing
(Figure~\ref{fig:ancillary} c \& g and and Figure~S\ref{fig:sup_ancil}
c).  These correlations support the hypothesis that the density
difference between the SoG and the east end of Juan de
Fuca Strait drives the estuarine exchange and that it is throttled by
high mixing due to the tides.

Lastly comparing the four-day low pass filtered Sand Head
winds to the baroclinic flux there is no clear correlation (Figure~\ref{fig:ancillary} d \& h and and Figure~S\ref{fig:sup_ancil}
d), but on the
short term there is an inverse relationship to the barotropic flux (Figure~\ref{fig:ancillary} h).
Given the strong relation between the winds and the four-day
low pass filtered sea surface height at Neah Bay, we see that the
barotropic flux may be driven by sea surface height fluctuations,
rather than directly by the wind.

The baroclinic flux through Boundary Pass given by equation~\ref{eq:theory}  agrees well with the results from Ariane
(Figure~\ref{fig:theoryfit}) replicating both the seasonal cycle
(which is mostly due to the density forcing $D_F$) and the fortnightly
cycle (which is mostly due to the $Ri$).  Here we have used a width of
10 km and a depth of 50 m and the proportionality constant is 0.0335
or about 1/30.  The low constant is expected as the fluid is
continuously stratified rather than two-layer.  For a gravity current
propagating into a continuously stratified fluid the velocity is lower
(missing citation).  In addition, our $Ri$ (on order of 2-15) is based on the
tidal flows themselves, not the resulting turbulent fluctuations.  The
implicit reduction from the flows to the fluctuations is also encapsulated in the constant.

One can then ask if adding any of the other forcing variables can
improve the fit.  Correlations of the residual to the wind (0.3), sea
surface height (0.3), freshwater flow (0.03) are all low.  The base
model explains 95.5\% of the variance.  Together these suggest that
the model does represent the baroclinic transport well.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/theory-fit-nw/theory-fit-nw}
\caption{{\label{fig:theoryfit} Comparison of the theoretical flux estimate (blue line, left
    panel) with the
    calculation using Ariane with SalishSeaCast currents (orange line,
    left panel).  Left panel: timeseries over the four years.  Right
    panel: histogram of the comparison with theory on the horizontal
    axis and the calculation on the vertical axis.  The black line is the
    one-to-one line. p-value $<$ 0.001, R$^2$ is 0.955.%
}}
\end{center}
\end{figure}

\subsection*{Properties of the Transport}

Ariane also allows us to ask what salinity the water had when it
started and ended its journey from/to Point Roberts to/from Victoria
Sill.  Binning by month and salinity shows clear seasonal patterns in
salinity of the transport (as well as the volume of the transport) (Figure~\ref{fig:FluxbySalt}).
We can clearly see flux of the densest water,  greater than 32 g kg$^{-1}$
is only significant in August and September.  Water that would be deep
water in the SoG ($>$ 31 g kg$^{-1}$) enters from May to November with
the strongest fluxes in August and September.  Very little deep water
leaves the SoG for Victoria Sill and that only happens in the winter.

Taking intermediate water to be between salinities of 30 and 31 g kg$^{-1}$, we see incoming
flux throughout the year with the strongest fluxes in the early part
of the summer (May-July).  Intermediate water also leaves the SoG
throughout the year with the strongest fluxes in early fall.

Upper water leaves throughout the year with the lowest salinities in
the summer.  The main outgoing flux is between 28.5 and 30 g
kg$^{-1}$.  Small amounts of upper water also come into the SoG,
primarly in late winter (Feb-Apr).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fluxbysalt/fluxbysalt}
\caption{{\label{fig:FluxbySalt} Histogram of volume flux into (left panel) and out of (right
  panel) Strait of Georgia separated by salinity class (vertical axis)
  and month (horizontal axis).  The salinities of the incoming waters
  are denser and cover less range of salinity than the outgoing
  waters.  Particularly in the summer, outgoing water can be quite
  brackish.  During the late summer early fall, fluxes are highest and
  the incoming water has the highest salinity.%
}}
\end{center}
\end{figure}

\subsection*{Geometry of the Flow}

Ariane also allows us to ask where, laterally and in depth, the water
leaves a transect and where it arrives at a transect.  In September,
during the deep water renewal season, we see strong inflow in the deep
SoG along the Roberts Bank line (Figure~\ref{fig:crossPR}) with
salinities above 31.4 g kg$^{-1}$.  There is
also less saline water (centered at 30.8 g kg$^{-1}$) incoming, on the mainland side of the transect.  This
cross-section is in the region where the flow in from Boundary Pass
moves to the right to hug the ``right-hand side'' of the SoG
(missing citation).  We also
see some incoming transport between the Gulf Islands.

The outgoing transport is surface focussed, as expected, with a much larger
contribution between the Gulf Islands.  The outgoing
transport is predominately on the west side of the transect, again as
expected.  Particularly in September, the outgoing flux is significant down to depths of at least
40~m (note the log scale of the flux).

During other times of the year (for example, March and December), the deep flux is not present and we
see only the intermediate salinity flux (Figure~\ref{fig:crossPR}).
Some of this flux is even deeper than 150~m as the salinities throughout the
transect are lower at this time of year.  The intermediate water flux
out of the SoG is deep on the western side of the transect.

Generally the incoming and outgoing flux do not overlap in space,
with the incoming flux deep on the west and intermediate depths on the
east and the outgoing flux shallow, especially on the east.  However,
there are regions of overlap, as during the month the locations of
incoming and outgoing parcels vary.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/PointRobertsFlux/PointRobertsFlux}
\caption{{\label{fig:crossPR} Monthly averaged flow, based on Lagrangian tracking, through the Point Roberts
  Cross-section.  Left Column: flow from Victoria Sill, Right Column:
  flow to Victoria Sill.  Green contours are month averaged salinity and are not
  equally spaced.  White cells have mean velocities lower than 2 mm s$^{-1}$.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/VictoriaSillFlux/VictoriaSillFlux}
\caption{{\label{fig:crossVS} Monthly averaged flow, based on Lagrangian tracking, through the Victoria Sill
  Cross-section.  Left Column: flow from Strait of Georgia, Right Column:
  flow to Strait of Georgia.  Green contours are month averaged salinity and are not
  equally spaced.  White cells have mean velocities lower than 2 mm s$^{-1}$.%
}}
\end{center}
\end{figure}

The location at Victoria Sill of the water transiting from the Point Roberts transect
to and from Victoria Sill shows a more consistent spatial pattern
(Figure~\ref{fig:crossVS}) but equally strong variations in salinity and, of
course, in flux.  Incoming water is deep with the strongest flux at
75~m above the two valleys in the sill. The densest water enters
through the northern valley and is seen through the months of June-August.

Outgoing water is shallow with the strongest flows above the two
valleys, with the northern dominant.  The freshest water leaving is
against the northern, Canadian, coast as expected due to rotational
effects.  The freshest water at this transect is seen in June and July.

\section*{Discussion}

First we will discuss our results in terms of previous studies of the
Salish Sea, comparing our flux estimates and describing the geometry
of the fluxes.  Then we will expand to considering the implications on
biogeochemical tracers.  At the end of the discussion we will return
to the general dynamical balance we have derived and use it to look
into the future.

\subsection*{Salish Sea}

\subsubsection*{Flux Comparison}

Flux estimates for transects near Point Roberts and transects near
Victoria Sill have been previously made using a expanded Knudsen's
relations mixing box based on observations (missing citation),
tidal averages based on an FVCOM model of the Salish Sea
(missing citation) and Total Exchange Flow based on a ROMS
model of the Salish Sea (missing citation).  Each of these are
different estimates of transport but are all similar in that they
measure transport across a transect; in contrast, the Lagrangian
transport calculation here measures the transport between transects
(that is, that component of the transport from one to the other).  The
reflux (missing citation) of flux back across the transect is
not included.  For the Victoria Sill flow toward Point Roberts transect,
the mixing box gives a reflux estimate of more than 78 mSv, or more
than 2/3 of the incoming flow (missing citation).

Ariane also tracks parcels that return and cross over their origin
transect (loops in Ariane terminology). The flux that returns back
across the Victoria Sill is 460 mSv and that back across the Point Roberts
transect is 280 mSv.  These estimates include, in addition to the
reflux, tidal sloshing (the movement of water without change back and
forth across the transect) which is
``removed'' in various ways by the alternative methods.  Both the mixing box
and the TEF methods should include the net impact of the tides whereas
tidal averaging excludes it.  We can
roughly split the loops into a tidal sloshing versus reflux using the
time length of the
loop, assuming times less than 24 hours are due to tidal sloshing.  The
longer loops, an estimate of the reflux, are 80~mSv for Victoria Sill
and 56~mSv for Point Roberts.

Gathering all these provisos together it makes best sense to compare
the four methods using the Point Roberts transect for the transect-based estimates.  The baroclinic fluxes are 72 mSv (TEF), 42 mSv
(mixing box), 31 mSv (Lagrangian,
including flux to Puget Sound) and 30 mSv (tidal average).  The order
is as expected, TEF includes all fluxes, the mixing box forces a
boundary between layers and thus misses some of the TEF, the Lagrangian method excludes the reflux, and the tidal
average excludes the net impact of the tides.

\subsubsection*{Flux Geometry}

The annual fluxes (Table~\ref{tab:meanfluxes}) show that 10\% of the flux north
through the transect into the SoG occurs in the Gulf
Islands and that increases to 25\% when considering the flux south.
It is not surprising, due to the influence of the Coriolis Force, that
the flux south is more constrained to the west side (the Gulf Island
side). However, the total contributions are higher than expected and
future observational campaigns should better survey this region.
On the other hand to generate the bathymetry for grid cells of 500~m, on average,
channels were widened, so these figures are probably upper bounds.

Within the SoG at the Point Roberts cross-section we also see the
impact of the Coriolis Force, flow north at mid-depth is on the east
side as described from observational studies ({\it e.g.} (missing citation)).  The
deepest flow is constrained to the deepest channel, which at this
cross-section is on the west side.  Observations have shown that this deepest flow, which enters
only when fluxes are largest, forms a rotationally constrained gravity
current in this region (missing citation) with height of
order 30~m (1.5 vertical grid cells for SalishSeaCast).

The surface southward flow at Point Roberts
stretches across the SoG with the strongest flux on the west side.
Intermediate southward flow is on the west side of the basin.  Again
this distribution is consistent with the impact of the Coriolis Force.

The fact that there is significant intermediate water flowing
southward has been suggested recently (missing citation) but was
not decernible in earlier, observational studies
(missing citation).  The classic view point in
(missing citation) suggests that the intermediate
water upwells into the surface layer to exit the Strait. However, our
results show that, in addition, a considerable outflow occurs at intermediate depths and
thus the intermediate layer has a smaller residence time.

The flux geometry above Victoria Sill also shows the influence of the
Coriolis Force with the outward flow on the north side and inward flow
more on the south side as expected from many previous studies
({\it e.g.} (missing citation)).  Also clear here is the impact of the two valleys in the
sill with much stronger inward and outward flow over the valleys.

When tracking the flows with Ariane, we use a closed box so we also
see the flux from the SoG to Puget Sound and the flux from Victoria
Sill to Puget Sound.  The flux to Puget Sound from Victoria Sill (32.9 mSv) and from Puget Sound
to Victoria Sill (37.7 mSv) is larger than the exchange flux with SoG.
 We need to be careful here though, as we are not making similar
 comparisons.  Our Puget Sound transects are at the mouth of Puget
 Sound, outside the strongest tidal mixing region, whereas our SoG transect is inside
 the strongest tidal mixing region.

There is also an exchange flow between Point Roberts and Puget Sound
with 4.2 mSv on average going south and 2.1 mSv going north.  Water
moves south continuously throughout the year with the strongest fluxes
in the summer.  The water is relatively fresh (average 28.4~g~kg$^{-1}$), particularly in the summer, when it
leaves the Point Roberts transect at average depths of 35~m but is
substantially saltier (average 30.4  g kg$^{-1}$)  by the time it
reaches the Puget Sound transect.  The flux going north is
intermittent, going to zero through part of the tidal cycle in the
summer.  The salinity changes little during transport north with
average 30.4 g kg$^{-1}$ when leaving Puget Sound and 30.2 g kg$^{-1}$ when reaching the Point
Roberts transect.  The drivers and pathways of exchange between the
SoG and Puget Sound are the subject of ongoing research
beyond this study.

\subsubsection*{Implications for Biology}

Plotting the mean monthly nitrate distribution on the flux
cross-sections (Figure~\ref{fig:nitrate}) we can see that the patterns
are much more complex than lower nitrate water fluxing out and high
nitrate water fluxing in.  In March, the water in and out seems to
have similar nitrate concentrations, and in June and December we see the
expected pattern.  However, in September, the highest nitrate water,
at least at mid-depth is being transported out of the SoG not in.  A thorough study of nitrate, carbon and alkalinity
transport is under way, including tracking the initial and final
concentration for each of the Ariane particles.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/PointRobertsNitrate/PointRobertsNitrate}
\caption{{\label{fig:nitrate} Monthly averaged flow, based on Lagrangian tracking, through the Point Roberts Cross-section
  with contours of monthly averaged nitrate in blue.  Left Column: flow from
  Victoria Sill. Right Column: flow to Victoria Sill.%
}}
\end{center}
\end{figure}

\subsection*{Flux Drivers}

The results clearly show that we can estimate the exchange flow across
a tidal mixing hotspot based on the laboratory models.  The flux is
well explained by the density difference across the mixing region and
a Richardson number that estimates the strength of the turbulent
mixing and its slowing of the exchange flow. Such a
balance is consistent with previous research that has shown
the role of tides in modulating the flux, usually incorporated through
the cube-root-mean-cubed tidal flow
(missing citation); (missing citation); (missing citation). Here we find a
dependence on the $Ri$ which is proportional to the root-mean-squared
tidal flow. The suggested
drivers of the exchange include the Fraser River flux ({\it e.g.}
(missing citation)) and upwelling on the offshore coast ({\it
  e.g.} (missing citation). These drivers affect the
density on the two sides of the mixing region which is encapsulated
here in the density difference.

Given our improved
dynamical formulation of the flux, one should be able to
extrapolate to other systems. We can also ask how and why the flux
through systems
may change interannually or due to climate change.
Looking at our system we can calculate the decrease in fluxes between 2017 and 2018
in three ways. Calculated directly from the Lagrangian tracking, it is 4\%;
calculated by the theory
averaging Equation~(\ref{eq:theory}) daily, it is 3\%; whereas averaging the density forcing and tidal
speed annually before estimating the mean, it is 4\%.
Using the third method, we can look at a longer
timeseries based on our long hindcast from 2007 to 2021
((missing citation), \url{salishsea.eos.ubc.ca/erddap}), we see a bit
more variation with the maximum difference between 2013 and 2010 being 12\%.
This variation is driven mostly by the density difference with
variations of its square almost as high (maximum between 2013 and 2014 of 11\%);
whereas, tidal variations (as the square root of the speed squared)
vary by at most 4\% (maximum between  2007 and 2017).

Thus as density differences vary between the Strait of Juan de Fuca
(primarily driven by upwelling/downwelling cycles on the open coast)
and the SoG (primarily driven by freshwater flux), the
annual flux of exchange will change.  As we expect similar to higher
fluxes of freshwater (missing citation) and generally higher
upwelling (missing citation), we
should expect stronger estuarine flows in the future, leading to
stronger flushing and lower residence times.

On the other hand, the peak densities and nutrients enter in the late
summer due to peak flows.  These peak flows are driven by the density
differences in the late summer, which we expect to {\em increase} due
to increasing strength of upwelling winds but {\em decrease} as the
late summer river flows decrease with the hydrograph moving earlier
(missing citation).  Thus, the densiest water entering the SoG may not change
strongly, as these two impacts compete.

\section*{Summary and Conclusions}

Based on an accurate numerical model of the Salish Sea we have used a
Lagrangian technique to investigate the flux through a tidal mixing
constriction. We mapped the fluxes at the two ends of the constriction,
showing that they are strongly laterally sheared, as well as sheared in
the vertical (Figure~\ref{fig:summary}).  Similarly the technique showed considerable advection of
intermediate water out of the inner basin (the Strait of Georgia) as
well as into it.

The Lagrangian method measured the actual through flux, neglecting flux
that is refluxed from one layer into another.  This transport flux is well described as a
gravity exchange through a mixing box. That is the flux is driven by
the density difference between the two sides of the mixing box and is
controlled by the mixing as measured by a Richardson number. This result is consistent
with drivers discussed previously, but gives a more consistent, accurate
formulation that should be transferable to other systems.

\subsection*{Acknowledgments}
The development of SalishSeaCast was made possible by Marine Environment Observation Prediction and Response Network (MEOPAR), which supported the second, fourth and fifth authors.  National Science and Engineering Council supported the second author through a Postdoctoral Fellowship and the third author through an Accelerator Grant.   Computing resources were provided by Compute Canada (now Digital Alliance), Ocean Networks Canada (ONC) and Advanced Research Computing (ARC) at University of British Columbia.  All grants were to the first author. Useful discussions on Lagrangian tracking and exchange flow were had with Becca Beutel, Camryn Stang, Jose Valenti and Rich Pawlowicz.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/estuarine-final-figure-flip/estuarine-final-figure-flip}
\caption{{\label{fig:summary} Sketch showing the yearly mean fluxes between Point Roberts
  (upper) and Victoria Sill (lower) transects.  The surface flux ($<
  30 $g kg$^{-1}$ at Point Roberts)
  southward is in green, the intermediate flux southward is in blue
  ($>  30 $g kg$^{-1}$ at Point Roberts).  The deep flux ($>  31.2 $g
  kg$^{-1}$ at Point Roberts) is in
  purple and the intermediate flux ($<  31.2 $g kg$^{-1}$ at Point Roberts)
  northward is in orange.  Note that the fluxes are dominated by
  surface water south and intermediate water north, with about one
  quarter as much intermediate water south and deep water north.  The
  impact of the Coriolis Force forcing flows to the right side of their
channels is clear. Not shown is the strong mixing between the
cross-sections.%
}}
\end{center}
\end{figure}

\clearpage

\section*{Supplementary Materials}\selectlanguage{english}
\begin{table}[bh]
\begin{tabular}{|ccc|cccc|c|}
\hline
{\bf From} & {\bf To} & {\bf Direction} & {\bf 2015} & {\bf
                                                            2016} &
                                                                    {\bf
                                                                    2017}
  & {\bf 2018} & {\bf Mean}
  \\
&&& (mSv) & (mSv) & (mSv) & (mSv) & (mSv) \\
\hline
Puget & Victoria & Backward & 36.5 & 36.2 & 37.3& 35.8 & 36.4\\
Puget & Point Roberts & Backward & 1.7 & 1.8 & 1.8 & 1.7 & 1.7\\
Puget & Gulf Islands & Backward & 0.2 & 0.2 & 0.2 & 0.2 & 0.2\\
Victoria & Puget & Forward & 32.1& 31.8 & 32.6 &  32.0 & 32.1\\
Point Robert & Puget & Forward & 3.5 & 3.7 & 3.5 & 3.3 & 3.5\\
Gulf Islands & Puget & Forward & 0.2  & 0.2 & 0.2 & 0.2 & 0.2\\
\hline
Discovery & Victoria & Backward & 1.3 & 1.3 & 1.4 & 1.2 & 1.3\\
Discovery & Point Roberts & Backward & 0.2  & 0.2 & 0.2 & 0.2 & 0.2\\
Discovery & Gulf Islands & Backward & 0.0 & 0.0 & 0.0& 0.0 & 0.0\\
Victoria& Discovery& Forward & 0.8& 0.7& 0.8 & 0.8 & 0.8\\
Point Roberts & Discovery & Forward & 0.5 & 0.5  & 0.5& 0.5 & 0.5\\
Gulf Islands & Discovery & Forward & 0.0 & 0.0 & 0.0& 0.0 & 0.0\\
\hline
\end{tabular}
\caption{{Annual Averaged Fluxes and Four Year Mean}}
%Transport-Analysis
\label{tab:sup_PSmeanfluxes}

\end{table}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/transports-supp/transports-supp}

\end{center}
\end{figure}

\begin{sidewaysfigure}
{}

\caption{{Transports calculated using Ariane.  Panel
    (a) and (b), deep water from Victoria Sill to Point Roberts
    and the Gulf Islands.  Panel (c) and (d), shallow water from Point
  Roberts and the Gulf Islands to Victoria Sill.  Flux is calculated
  both by tracking particles forward (time is when they start) and by
tracking particles backward (time is when they arrived). Panel (b) and
(d) show the time scale shifted to bring the forward and backward time
series together based on the time it took the particles to travel
between the transects (see text).  The red line is the mean flux, averaged between the
two shifted time series.}}
\label{fig:sup_transports} 

\end{sidewaysfigure}

\begin{sidewaysfigure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Ancillary-sup/Ancillary-sup}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{./static.png}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fluxbysaltbyyear/fluxbysaltbyyear}
\caption{{\label{fig:sup_FluxbySalt} Histogram of volume flux into (left panels) and out of (right
  panels) Strait of Georgia separated by salinity class (vertical axis)
  and month (horizontal axis).%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/PointRobertsFlux2015-allmonths/PointRobertsFlux2015-allmonths}
\caption{{\label{fig:sup_crossPR} Monthly averaged flow, based on Lagrangian tracking, for 2015 through the Point Roberts
  Cross-section.  Left Column of each pair: flow from Victoria Sill,
  Right Column of each pair:
  flow to Victoria Sill.  Green contours are salinity and are not
  equally spaced.  White cells have mean velocities lower than 2~mm~s$^{-1}$.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/VictoriaSill2015-allmonths/VictoriaSill2015-allmonths}
\caption{{\label{fig:sup_crossVS} Monthly averaged flow for 2015, based on Lagrangian tracking, through the Victoria Sill
  Cross-section.  Left Column of each pair: flow from Strait of
  Georgia, Right Column of each pair:
  flow to Strait of Georgia.  Green contours are salinity and are not
  equally spaced.  White cells have mean velocities lower than 2~mm~s$^{-1}$.%
}}
\end{center}
\end{figure}

\selectlanguage{english}
\FloatBarrier
\section*{References}\sloppy
\end{document}

