may have mass contributions from external water sources
or sinks (evaporation, precipitation, and subsurface water
springs from aquifers), and two vertical sections enclosing
the estuarine water body.
A simple integrated equation of salt conservation, taking into account only the advection process on salt mixing,
may be easily obtained applying the divergence operator to
the mass flux vector S Á r v
! S Á r v
!
h
i
¼ M Á L
À2 T
À1 .
Under the hypothesis that the salinity (density) and the
velocity field in the estuarine domain are in steady state
represented by its time mean value during tidal cycles,
the divergence of the salt mass flux is zero:
∇ • Sr v
! ¼ 0
ð6Þ
and as in (5)
Z
V
∇ • Sr v
!
dV ¼
Z
A
Sr v
! • n
!
dA ¼ 0
ð7Þ
the salt transport (MT
À1
, kg s
À1 ) over a closed area is zero.
With simplified geometry, (5) and (7) may be
transformed in a system with two unknowns. They were
used in the Knudsen hydrographic theorem applied for the
first time around 1,900, enabling mean velocities to be calculated in cross-section areas under steady-state conditions,
with known mean salinities in a highly stratified estuary
(salt wedge) and with the fresh water discharge (Q f ) as the
main forcing mechanism. In this type of estuary, the salt
transport is driven by river discharge and the vertical turbulent mixing is absent. Let A s (u s ) and A i (u i ) be the superior
and inferior cross-section areas (mean velocities) limited by
the halocline, and S s and S i the mean salinities, respectively.
Disregarding the mass inflow and outflow across the
free surface and the bottom, the equation system of (5)
and (7) may be applied, taking into account that
v
! • n
!
6 ¼ 0 only on the areas A s and A i , then:
Z
A
v
! • n
!
dA ¼ u s A s À u i A i À Q f ¼ 0;
ð8aÞ
Z
A
Sr v
! • n
!
dA ¼ S s r s v s A s À S i r i v i A i
¼ 0 ð8bÞ
Disregarding the density differences in the upper and
lower layers (r s % r i ) in the (8b), this system may be
resolved for the mean current velocities (u s , u i ) and/or
the volume transports (Q s , Q i ):
u s ¼
S i Q f
A s S i À S s
ð
Þ
,
or
Q s ¼
S i Q f
S i À S s
ð
Þ
; ð9aÞ
u i ¼
S i Q f
A i S i À S s
ð
Þ
,
or
Q i ¼
S s Q f
S i À S s
ð
Þ
: ð9bÞ
An application of this result may be found in Miranda
et al. (2012) using the following experimental data of the
Fraser River, according to Geyer (1986): discharge
Q f ¼ 3,000 m
3
s
À1
; sections geometry A s ¼ 3,750 m
2
and A i ¼ 4,500 m
2
, and salinities S s ¼ 14.0 psu and
S i ¼ 30.0 psu. Then, the theoretical mean velocities
and volume transports are u s ¼ 1.5 ms
À1
, u i ¼ 0.6 ms
À1
,
Q s ¼ 5,525 m
3
s
À1
, and Q i ¼ À2,525 m
3
s
À1
, respectively.
These results show that the transport volumes are in
balance with the fresh water discharge.
Consider an estuary with a surface area (A) delimited
by the bottom and the surface and two vertical sections
A 1 and A 2 at the river zone (RZ, where S ¼ 0) and mixing
zone (MZ), respectively. It follows from (8a), the mean
longitudinal velocity under steady-state conditions across
the area A 2 , which is given by: À u f Á A 1 + u 2 Á A 2 ¼ À Q f +
u 2 Á A 2 ¼ 0, and thus u 2 ¼ u f ¼ Q f /A 2 is the fresh water
velocity driven by the river discharge (Q f ).
From the continuity equation (3), the conservation salt
principle due only to the advective process is:
@ rS
ð Þ
@t
þ ∇ • rS v
!
¼ 0
ð10aÞ
and
@S
@t
þ v
! • ∇S ¼ 0
ð10bÞ
However, the local variation
@S
@t
À Á
also depends on
the turbulent salt-diffusion flux (f S ), which is simulated
by Fick’s law f S ¼ ÀD
@S
@n
À Á
Â
Ã
(D is the dynamic diffusion
coefficient ([D] ¼ MÁL
À2 T
À1
) and
@S
@n is the directional
salinity gradient). The composition of this partial salt flux
with (10b) takes the expression of the salt conservation
equation (Sverdrup et al., 1942; Pritchard, 1958):
@S
@t
þ u
@S
@x
þ v
@S
@y
þ w
@S
@z
¼
@
@x
K x
@S
@x
þ
@
@y
K y
@S
@y
þ
@
@z
K z
@S
@z
À S sinks þ S sources
ð11Þ
and the local salinity variation
@S
@t
À Á
is determined by the
advection and diffusion (small-scale motion) processes,
and sources and sinks of salt (precipitation, evaporation,
bottom springs and sinks). In this equation, the Fickian
coefficients are, according to Osborne Reynolds in 1884,
parameterized in terms of the small-scale velocity (u
0 ,v
0
,w
0 )
and salinity fluctuations (S
0 ):
K x ¼ À
< u
0 S
0
>
@S
@x
; K x ¼ À
< v
0 S
0
>
@S
@y
; K x ¼ À
< w
0 S
0
>
@S
@z
;
ð12Þ
ESTUARINE CIRCULATION
251
or sinks (evaporation, precipitation, and subsurface water
springs from aquifers), and two vertical sections enclosing
the estuarine water body.
A simple integrated equation of salt conservation, taking into account only the advection process on salt mixing,
may be easily obtained applying the divergence operator to
the mass flux vector S Á r v
! S Á r v
!
h
i
¼ M Á L
À2 T
À1 .
Under the hypothesis that the salinity (density) and the
velocity field in the estuarine domain are in steady state
represented by its time mean value during tidal cycles,
the divergence of the salt mass flux is zero:
∇ • Sr v
! ¼ 0
ð6Þ
and as in (5)
Z
V
∇ • Sr v
!
dV ¼
Z
A
Sr v
! • n
!
dA ¼ 0
ð7Þ
the salt transport (MT
À1
, kg s
À1 ) over a closed area is zero.
With simplified geometry, (5) and (7) may be
transformed in a system with two unknowns. They were
used in the Knudsen hydrographic theorem applied for the
first time around 1,900, enabling mean velocities to be calculated in cross-section areas under steady-state conditions,
with known mean salinities in a highly stratified estuary
(salt wedge) and with the fresh water discharge (Q f ) as the
main forcing mechanism. In this type of estuary, the salt
transport is driven by river discharge and the vertical turbulent mixing is absent. Let A s (u s ) and A i (u i ) be the superior
and inferior cross-section areas (mean velocities) limited by
the halocline, and S s and S i the mean salinities, respectively.
Disregarding the mass inflow and outflow across the
free surface and the bottom, the equation system of (5)
and (7) may be applied, taking into account that
v
! • n
!
6 ¼ 0 only on the areas A s and A i , then:
Z
A
v
! • n
!
dA ¼ u s A s À u i A i À Q f ¼ 0;
ð8aÞ
Z
A
Sr v
! • n
!
dA ¼ S s r s v s A s À S i r i v i A i
¼ 0 ð8bÞ
Disregarding the density differences in the upper and
lower layers (r s % r i ) in the (8b), this system may be
resolved for the mean current velocities (u s , u i ) and/or
the volume transports (Q s , Q i ):
u s ¼
S i Q f
A s S i À S s
ð
Þ
,
or
Q s ¼
S i Q f
S i À S s
ð
Þ
; ð9aÞ
u i ¼
S i Q f
A i S i À S s
ð
Þ
,
or
Q i ¼
S s Q f
S i À S s
ð
Þ
: ð9bÞ
An application of this result may be found in Miranda
et al. (2012) using the following experimental data of the
Fraser River, according to Geyer (1986): discharge
Q f ¼ 3,000 m
3
s
À1
; sections geometry A s ¼ 3,750 m
2
and A i ¼ 4,500 m
2
, and salinities S s ¼ 14.0 psu and
S i ¼ 30.0 psu. Then, the theoretical mean velocities
and volume transports are u s ¼ 1.5 ms
À1
, u i ¼ 0.6 ms
À1
,
Q s ¼ 5,525 m
3
s
À1
, and Q i ¼ À2,525 m
3
s
À1
, respectively.
These results show that the transport volumes are in
balance with the fresh water discharge.
Consider an estuary with a surface area (A) delimited
by the bottom and the surface and two vertical sections
A 1 and A 2 at the river zone (RZ, where S ¼ 0) and mixing
zone (MZ), respectively. It follows from (8a), the mean
longitudinal velocity under steady-state conditions across
the area A 2 , which is given by: À u f Á A 1 + u 2 Á A 2 ¼ À Q f +
u 2 Á A 2 ¼ 0, and thus u 2 ¼ u f ¼ Q f /A 2 is the fresh water
velocity driven by the river discharge (Q f ).
From the continuity equation (3), the conservation salt
principle due only to the advective process is:
@ rS
ð Þ
@t
þ ∇ • rS v
!
¼ 0
ð10aÞ
and
@S
@t
þ v
! • ∇S ¼ 0
ð10bÞ
However, the local variation
@S
@t
À Á
also depends on
the turbulent salt-diffusion flux (f S ), which is simulated
by Fick’s law f S ¼ ÀD
@S
@n
À Á
Â
Ã
(D is the dynamic diffusion
coefficient ([D] ¼ MÁL
À2 T
À1
) and
@S
@n is the directional
salinity gradient). The composition of this partial salt flux
with (10b) takes the expression of the salt conservation
equation (Sverdrup et al., 1942; Pritchard, 1958):
@S
@t
þ u
@S
@x
þ v
@S
@y
þ w
@S
@z
¼
@
@x
K x
@S
@x
þ
@
@y
K y
@S
@y
þ
@
@z
K z
@S
@z
À S sinks þ S sources
ð11Þ
and the local salinity variation
@S
@t
À Á
is determined by the
advection and diffusion (small-scale motion) processes,
and sources and sinks of salt (precipitation, evaporation,
bottom springs and sinks). In this equation, the Fickian
coefficients are, according to Osborne Reynolds in 1884,
parameterized in terms of the small-scale velocity (u
0 ,v
0
,w
0 )
and salinity fluctuations (S
0 ):
K x ¼ À
< u
0 S
0
>
@S
@x
; K x ¼ À
< v
0 S
0
>
@S
@y
; K x ¼ À
< w
0 S
0
>
@S
@z
;
ð12Þ
ESTUARINE CIRCULATION
251