1. Overbar ū represents a sectional average at a given time
Salt Flux
Through a relatively straightforward calculation of salt flux we can learn about the relevant mechanisms responsible for transport of solutes. y z
u, S a
Definitions
Overbar ū represents a sectional average at a given time
prime u’ denotes deviations from that sectional average
brackets <u> indicate average over a tidal cycle
tilde represents intratidal variations
spatial
context
temporal
context

2. Salt Flux Calculations
The salinity and axial velocity at any point (and at any given time) on a cross section of an estuary may each be represented as the sum of a cross-sectional average plus the deviation (in space) from that average: y z A
The integral over the cross-section A of the product yields the rate of salt transport due to “mean flow”
= “mean flow”
The integral over the cross-section A of the product u’ S’ yields the instantaneous rate of salt transport due to “shear dispersion”
= “shear dispersion”
The shear dispersion related to spatial variations through the water column is the vertical shear dispersion
The shear dispersion related to variations across the width of the estuary is the transverse shear dispersion

3. Shear Dispersion x z
Vertical Shear Dispersion x y
Horizontal Shear Dispersion
Shear Dispersion ~ Spatial covariance between u and S

4. Transport Calculation
y, j
z, i
Ai j
area of each element
Make:
ui = transverse mean at depth i
ui t = transverse sum at depth i
Ai t = area of transverse strip at depth i
Ai t
Water transport Qi j = Ai j ui j
Salt transport Fi j = Ai j ui j Si j

5. The transport through each transverse strip is then given by:
Qi t = Ai t ui
Fi t = Ai t ui Si
y, j
z, i
Ai j
area of each element
Ai t
Av j
Then make:
uj = vertical or depth mean at strip j
uv j = vertical sum of strip j
Av j = area of vertical strip j
Therefore, the transport through each vertical strip is given by:
Qv j = Av j uj
Fv j = Av j uj Sj
And the rates of transport through the cross-section are represented as
Q v t and F v t

6. The vertical and transverse deviations of u i and u j are:
y, j
z, i
Ai j
area of each element
Ai t
Av j
The vertical and transverse deviations of u i and u j are:
u i’ = u i - ū
u j’ = u j - ū
ū is the sectional mean throughout A
The “interaction” deviation is:
u i j* = u i j - ( ū + u i’ + u j’ )
The rate of salt transport through the entire cross section is:

7. Qi j = Ai j ui j
Fi j = Ai j ui j Si j
u i’ = u i - ū
u j’ = u j - ū
u i j* = u i j - ( ū + u i’ + u j’ )
and using
and
That is the spatial representation at any given time. Now let’s look at time variations. We define a tidal oscillation as with zero average.

8. When including the temporal context to the Salt Flux calculation
“tidal pumping” arises

9. Tidal pumping arises as the flood water mixes with relatively fresher water. A portion of that mixed water leaves the estuary on ebb. Then, fresher water leaves the estuary during ebb and saltier water enters the estuary during flood. This leads to down-estuary (seaward) pumping of fresher water, or equivalently, up-estuary pumping of salt.
Tidal Pumping ~ Temporal covariance between u and S

10. Situation when Tidal pumping is most effective:
time
flood
time
flood
Perfect covariance between and

11. The residual rate of transport of salt through the cross-section is:
This relationship tells us the important mechanisms responsible for salt transport.
The same relationship applies for sediment transport (e.g. Uncles et al, 1985, Estuaries, 8(3), ).
It has also been used for calculations of seston transport (Pino et al., 1994, ECSS, 38, ).
Most recent reference on the approach: Jay et al., 1997, Estuaries, 20(2),

12. Example of Salt Flux by Tidal Pumping Seno Ballena Strait of Magellan
Antarctic expeditions leaving through the Strait of Magellan sail by this fjord
So the fjord is off Magellan Strait
Seno Ballena

13. Seno Ballena Glacier typical depths > 200 m 2 km Axial section
Here are the different sampling trajectories
The white, yellow, blue and red were trajectories repeated at least for 12 hours
The orange trajectory was an exploratory one that was done in Dec 2004 because we couldn’t cross the bar with the drafty boat we had the previous year
Note that the only passage over the sill through the region marked by the trajectories.
Across the bar, the depths are too shallow to navigate.
East of the sill, the depths are > 200 m
West of the sill…
The bathymetry is shallower and irregular
Note the larger slope of the bar on the east side and the sill practically blocking two basins
From the sill to the Glacier there are approximately 9 km
Glacier

14. hydrography suggests: blocking of landward transport of salt by sill
head
11 CTD stations along orange trajectory
hydrography suggests:
blocking of landward transport of salt by sill
- tidal pumping
mouth
Here we see the upper 20 m, where the action takes place, illustrating a very thin buoyant layer (3 m thick) of fresh and cold water
This layer is well contained west of the sill
Heavy water is blocked by the sill

15. Total salt flux <uS> continuous line;
Salt flux produced by mean flow <u><S> as dashed line;
Tidal pumping salt flux <u’S’> as dotted line.