Salt Flux
Through a relatively straightforward calculation of salt flux we can
learn about the relevant mechanisms responsible for transport of
solutes.
z
u, S
a
a u S
y
Definitions
spatial Overbar ū represents a sectional average at a given time
context
prime u’ denotes deviations from that sectional average
brackets <u> indicate average over a tidal cycle
temporal
context tilde u~ represents intratidal variations
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:
u  u  u'
z
S  S  S'
A
y
The integral over the cross-section A of the product u S yields the rate of
salt transport due to “mean flow”  u  S  dA = “mean flow”
A
The integral over the cross-section A of the product u’ S’ yields the
instantaneous rate of salt transport due to “shear dispersion”
 u 'S 'dA = “shear dispersion”
A
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
z
y
x
x
Vertical Shear Dispersion
Horizontal Shear Dispersion
Shear Dispersion ~ Spatial covariance between u and S
Transport Calculation
z, i
Ai j
area of
each
element
Ai t
y, j
Water transport
Salt transport
Qi j = Ai j ui j
Fi j = Ai j ui j Si j
Make:
ui = transverse mean at depth i
ui t = transverse sum at depth i
Ai t = area of transverse strip at depth i
The transport through each
transverse strip is then given
by:
z, i
Ai j
Ai t
area of
each
element
Qi t = Ai t ui
Fi t = Ai t ui Si
Then make:
uj = vertical or depth mean at strip j
uv j = vertical sum of strip j
Av j = area of vertical strip j
Av j
y, 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
z, i
The vertical and transverse
deviations of u i and u j are:
Ai j
Ai t
area of
each
element
u i’ = u i - ū
u j’ = u j - ū
ū is the sectional mean
throughout A
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:
Fvt  Avt uij Sij
y, j
Fvt  Avt uij Sij
Qi j = Ai j ui j
Fi j = Ai j ui j Si j
and using
u i’ = u i - ū
u j’ = u j - ū
u i j* = u i j - ( ū + u i’ + u j’ )
Fvt  Qvt S  Avt u i ' Si '  Avt u j ' S j '  FS*
and
FS*  Avt ui ' S j '  Avt u j ' Si '  Avt ui ' S ij*  Avt u j ' S ij*  Avt u ij* Si '  Avt u ij* S j '  Avt u ij* S ij*
That is the spatial representation at any given time. Now let’s look at time
~ with zero average.
variations. We define a tidal oscillation as Q
~
 Qvt  Qvt  Qvt
~
S  S S
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:
flood
S
time
flood
u
time
Perfect covariance between u~ and S~
The residual rate of transport of salt through the cross-section is:
Fvt  FL  FTP  FvS  FSt 




residual
flow
tidal
pumping
vertical
shear
dispersion
transverse
shear
dispersion
FS*

interactio ns
between vertical
and transverse
deviations
FL  Qvt S
~ ~
FTP  Qvt S
FvS  Avt u i ' Si '
FSt  Avt u j ' S j '
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
Strait of Magellan
Seno Ballena
2 km
Axial section
Glacier
head
11 CTD stations along orange trajectory
hydrography suggests:
- blocking of landward
transport of salt by sill
- tidal pumping
mouth
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.