Salt Flux

Through a relatively straightforward calculation of salt flux we can learn about the relevant mechanisms responsible for transport of solutes.

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 variationsu~

spatialspatialcontextcontext

temporaltemporalcontextcontext

y

z u, S

Sua

a

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:

'

'

SSS

uuu

The integral over the cross-section A of the product u’ S’ yields the instantaneous rate of salt transport due to “shear dispersion”

A

dASu '' = “shear dispersion”

SuThe integral over the cross-section A of the product yields the rate of salt transport due to “mean flow”

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

y

z

A

A

dASu = “mean flow”

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

Water transport Qi j = Ai j ui j

Salt transport Fi j = Ai j ui j Si j

Make:ui = transverse mean at depth iui t = transverse sum at depth iAi t = area of transverse strip at depth i

Ai t

y, j

z, i

Ai jarea of each element

Ai t

The transport through each transverse strip is then given by:

Qi t = Ai t ui

Fi t = Ai t ui Si

Then make:uj = vertical or depth mean at strip juv j = vertical sum of strip jAv j = area of vertical strip j

Av 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:

u i’ = u i - ūu j’ = u j - ū

ū is the sectional mean throughout A

y, j

z, i

Ai jarea of each element

Ai t

Av j

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:

ijijvtvt SuAF

*'''' Sjjvtiivtvtvt FSuASuASQF

******* ''''''''ijijijijijij

SuASuASuASuASuASuASuAF vtjvtivtjvtivtijvtjivtS

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.Q

~

SSS

QQQ vtvtvt

~

~

Qi j = Ai j ui j

Fi j = Ai j ui j Si j

u i’ = u i - ūu j’ = u j - ū

ijijvtvt SuAF

u i j* = u i j - ( ū + u i’ + u j’ )

and using

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:

-1 .5

-1

-0 .5

0

0 .5

1

1 .5

-0 .7 1 6 8 0 .2 8 3 2 1 .2 8 3 2 2 .2 8 3 2 3 .2 8 3 2 4 .2 8 3 2 5 .2 8 3 2 6 .2 8 3 2

S

time

-1 .5

-1

-0 .5

0

0 .5

1

1 .5

-0 .7 1 6 8 0 .2 8 3 2 1 .2 8 3 2 2 .2 8 3 2 3 .2 8 3 2 4 .2 8 3 2 5 .2 8 3 2 6 .2 8 3 2

u

time

flo

od

flo

od

Perfect covariance between andu~ S~

The residual rate of transport of salt through the cross-section is:

deviationstransverse and

vertical betweennsinteractio

*

dispersionsheartransverse

dispersionshearvertical

pumpingtidal

flowresidual

SFFFFFF StvSTPLvt

''

''

~~

jjvtSt

iivtvS

vtTP

vtL

SuAF

SuAF

SQF

SQF

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.

Seno Ballena

Strait of Magellan

Example of Salt Fluxby Tidal Pumping

Seno Ballena

Glacier

2 km2 km

typical depths > 200 m

Axial section

11 CTD stations along orange trajectory

head

mouthhydrography suggests:

- blocking of landward transport of salt by sill

- tidal pumping

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.