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

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

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

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

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

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:

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.

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

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

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

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), 256-269). It has also been used for calculations of seston transport (Pino et al., 1994, ECSS, 38, 491-505). Most recent reference on the approach: Jay et al., 1997, Estuaries, 20(2), 262-280.

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

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

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

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.