Estuary Classification Revisited A G A. L

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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 43
Estuary Classification Revisited
ANIRBAN GUHA AND GREGORY A. LAWRENCE
Civil Engineering Department, and Institute of Applied Mathematics, The University of British Columbia,
Vancouver, British Columbia, Canada
(Manuscript received 19 July 2012, in final form 31 March 2013)
ABSTRACT
Studies over a period of several decades have resulted in a relatively simple set of equations describing the
tidally and width-averaged balances of momentum and salt in a rectangular estuary. The authors rewrite these
equations in a fully nondimensional form that yields two nondimensional variables: (i) the estuarine Froude
number and (ii) a modified tidal Froude number. The latter is the product of the tidal Froude number and the
square root of the estuarine aspect ratio. These two variables are used to define a prognostic estuary classification scheme, which compares favorably with published estuarine data.
1. Introduction
Since the introduction of the stratification–circulation
diagram by Hansen and Rattray (1966), numerous estuarine classification schemes have been proposed. The
reader might ask—why revisit this topic? Our motivation
for pursuing a new classification scheme stems from notable recent advances in estuarine physics, many of which
are reviewed in MacCready and Geyer (2010). These advances led us to hypothesize that there might be a simple
means to determine the conditions under which a sufficiently well behaved estuary will be well mixed, partially
mixed, or highly stratified. We start by outlining the classical tidally averaged model as presented by MacCready
and Geyer (2010). We then rewrite the equations of this
model in nondimensional form. Using this new set of
equations we develop our classification scheme, and then
compare its predictions with field observations.
exchange flow, and the tides determines the estuarine
velocity and salinity structure.
We consider an idealized rectangular estuary of depth
H and width B. The origin of the coordinate system is
at the free surface at the mouth of the estuary with the
horizontal (i.e., x) axis pointing seawards and the vertical
(i.e., z) axis pointing upward. Therefore, both the horizontal and vertical distances within the estuary are negative quantities. To obtain the width- and tidally averaged
horizontal velocity u, and salinity s distribution in the
estuary, these quantities are first decomposed into depthaveraged (overbar) and depth-varying (prime) components: u 5 u(x, t) 1 u0 (x, z, t) and s 5 s(x, t) 1 s0 (x, z, t).
The quantity u 5 QR /A is the cross-section-averaged
river velocity, where QR is the mean river flow rate, and
A 5 BH. The solution for both partially and well-mixed
estuaries was given by Hansen and Rattray (1965) [for
a recent review, see MacCready and Geyer (2010)]:
u 5 u 1 u0 5 uP1 1 uE P2
2. Classical tidally averaged model
The physics of estuarine circulation is governed by the
competing influences of river and oceanic flows. While
the former adds freshwater, the latter adds denser saltwater, which moves landward because of the combined
effect of tides and gravitational circulation (or exchange
flow). The complicated balance between the river, the
Corresponding author address: Anirban Guha, Civil Engineering
Department, The University of British Columbia, 2002–6250 Applied Science Lane, Vancouver, BC V6T 1Z4, Canada.
E-mail: [email protected]
DOI: 10.1175/JPO-D-12-0129.1
Ó 2013 American Meteorological Society
s 5 s 1 s0 5 s 1
where
and
H2
s (uP3 1 uE P4 ) ,
KS x
(1)
(2)
3 3
P1 5 2 j2 ,
2 2
P2 5 1 2 9j2 2 8j3 ,
7
1
1
1 j2 2 j 4 , and
P3 5 2
120 4
8
1 1 2 3 4 2 5
P4 5 2 1 j 2 j 2 j .
12 2
4
5
(3)
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In Eq. (3), j 5 z/H 2 [21, 0] is the normalized vertical
coordinate. The subscript x implies ›/›x, where x is
dimensional. The vertical eddy diffusivity is KS . For
exchange-dominated estuaries, an important parameter
is the exchange velocity scale:
uE 5 c2 H 2 Sx /(48KM ) .
ð
d
S dx 5 2u0 S0 1 KH S x 2uS ,
dt
|fflfflfflfflffl{zfflfflfflfflffl}
|{z}
|fflfflffl{zfflfflffl} |ffl{zffl}
accumulation
exchange
tidal
where
E2
E1
T
LE 5 0:019uH 2 /KS ,
1
3
The different terms in Eq. (6) are as follows: R is the
river term; T is the tidal term; and E1, E2, and E3 are the
different components of the exchange term. Hansen and
Rattray (1965) presented Eq. (6) in a slightly different
form, and MacCready (2004, 2007) introduced the
length scales in Eq. (7).
The length scales in Eq. (7) depend upon the mixing
coefficients: KH , KS , and KM . Making use of an extensive study of Willapa Bay, Banas et al. (2004) proposed
the following:
(9)
E3
(10)
T
Chatwin (1976) further reduced Eq. (10) to two simple
cases with analytical solutions: the exchange-dominated
case (T / 0) and the tidally dominated case (E3 / 0).
While these approximations have been widely used, there
does not appear to have been any serious attempt to determine the conditions under which they are applicable.
3. Nondimensional tidally averaged model
In this section, we rewrite the governing Eqs. (1), (2),
and (6) in nondimensional form in anticipation of:
(i) revealing the important nondimensional parameters
governing the problem and (ii) facilitating comparison
of the relative magnitude of each of the terms in Eq. (6).
Defining X 5 x/LE3 , Eq. (6) can be rewritten as
and
(7)
KS 5 KM /Sc,
S 5 (LE S x )3 1 LH S x .
3
|{z}
|fflfflfflfflfflffl{zfflfflfflfflffl
ffl}
|fflffl{zfflffl}
2
2 1/3
LE 5 0:024(c/u)1/3 cH 2 /(KS KM
) .
and
where a0 5 0.028, CD 5 0.0026, and Sc 5 2.2 is a Schmidt
number. We will use Eqs. (8) and (9) in the development
of a nondimensional set of equations.
While the governing Eqs. (1), (2), and (6) are elegant
representations of the problem of estuarine circulation,
they are sufficiently complicated that simplifications
have been sought after. Numerous investigators, including
Hansen and Rattray (1965), Chatwin (1976), Monismith
et al. (2002), MacCready (2004), and MacCready and
Geyer (2010) have assumed u uE , which yields
R
LH 5 KH /u,
LE 5 0:031cH 2 /(KS KM )1/2 ,
KM 5 a0 CD uT H
river
S 5 (LE S x)3 1 (LE S x)2 1 LE S x 1 LH S x , (6)
3
2
1
|{z} |fflfflfflfflfflffl{zfflfflfflfflffl
ffl}
|fflfflfflfflfflffl{zfflfflfflfflffl
ffl} |fflfflffl{zfflffl
ffl}
|fflffl{zfflffl}
(8)
where a1 5 0.035, and uT is the amplitude of the depthaveraged tidal velocity. Based on field studies and modeling of the Hudson River estuary, Ralston et al. (2008)
obtained
(5)
where KH is the horizontal diffusivity. This equation
physically implies that the temporal salt accumulation
in an estuary is due to the competition between salt
addition and removal processes. While exchange (note
that u0 S0 is negative) and tidal processes add salt, river
inflow removes it. At steady state, Eq. (5) can be rewritten as
E3
KH 5 a1 uT B,
(4)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Here, c 5 gbsocn H is twice the speed of the fastest
internal wave that can be supported in an estuary
(MacCready and Geyer 2010). The vertical eddy viscosity is KM and b ffi 7.7 3 10 24 psu21 . The nondimensional salinity is defined as S 5 s/socn, where socn
is the ocean salinity. Equations (1) and (2) were derived
under the assumption that the density field is governed
by the linear equation of state: r 5 r0(1 1 bs), where
r0 is the density of freshwater. The details of the derivation are well documented in MacCready (1999, 2004).
The salt balance is given by
R
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GUHA AND LAWRENCE
S
3
5SX
where
1
!2
LE
2
LE
3
2
SX
1
!
LE
1
LE
3
!
LH
SX 1
SX,
LE
(11)
3
!2
0:031 2 1/3 2/3
Sc FR 5 2:17FR2/3 ,
0:024
LE
3
LE
0:019
1
5
Sc2/3 FR4/3 5 1:34FR4/3 , and
0:024
LE
3
LH
a 0 a 1 CD
Sc21/3 (B/H)FT2 FR22/3 .
5
LE
0:024
LE
5
2
3
(12)
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The velocity u and uT have been nondimensionalized
by c to obtain the densimetric estuarine Froude number
FR 5 u/c and the tidal Froude number FT 5 uT/c.
Substituting Eq. (12) into (11) yields
3
2
f 2 F 22/3 S ,
S 5 S X 1 C1 FR2/3 S X 1 C2 FR4/3 S X 1 C3 F
R
T
X
|{z} |{z} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
R
E3
E2
E1
T
(13)
3ffiffiffiffiffiffiffiffiffi
1025
where C1 5 2.17, C2 5 1.34, C3 5 8.16 p
ffi , and the
f
modified tidal Froude number FT 5 FT B/H .
Like the salt balance equation, the momentum and
salinity equations [Eqs. (1) and (2)] can also be expressed
in nondimensional form as follows:
U 5 C4 FR1/3 S X P2 1 FR P1
and
2
S 5 S 1 C5 FR2/3 S X P4 1 C6 FR4/3 S X P3 .
(14)
(15)
The constant C4 5 0.667, C5 5 47.0, and C6 5 70.5. In
Eq. (14), the quantity U 5 u/c is the nondimensional
horizontal velocity (not to be confused with FR, which
is u/c). Equations (13)–(15) are the nondimensional
governing equations for our idealized estuary.
Equation (13) poses a nonlinear initial value problem,
which can only be solved numerically. For that, the
conditions at the estuary mouth have to be determined.
One such condition is S(0, 21) 5 1; meaning that the
salinity at the bed of the estuary at its mouth has to be
the same as the ocean salinity. Substituting Eq. (15) into
(13) and making use of this condition, we obtain
(S X j0 )3 1 C7 FR2/3 (S X j0 )2
f2 F 22/3 )S j 5 1,
1 (C8 FR4/3 1 C3 F
T R
X 0
(16)
where C7 5 5.31 and C8 5 6.04. Equation (16) is actually
the nondimensional version of Eq. (19) of MacCready
(2004). Being a cubic equation, it can be solved analytically to evaluate the salinity gradient at the estuary
mouth S X j0 . Additionally, Eq. (16) indicates that S X j0 is
fT . The variation of S X j with
only a function of FR and F
0
these two Froude numbers is depicted in Fig. 1. The
figure shows that 0 , S X j0 , 1 over the entire parameter
space.
4. Estuary classification
Our goal is to develop a simple classification scheme
that distinguishes between well-mixed, partially mixed,
and highly stratified estuaries. A relevant parameter
for classifying estuaries is the nondimensional salinity
FIG. 1. Variation of the horizontal gradient of depth-averaged
salinity at the estuary mouth (i.e., S X j0 ) with the estuary control
fT . The solid lines represent isocontours of S X j .
variables—FR and F
0
stratification at the estuary mouth F0 . It is defined as
follows:
F0 5 1 2 S(0, 0).
(17)
This parameter ranges between 0 and 1. While the lower
limit implies a very well-mixed estuary, the upper limit
indicates the transition to salt wedge. Substituting Eq.
(15) into (17) yields:
F0 5 C9 FR2/3 (S X j0 )2 1 C10 FR4/3 S X j0 ,
(18)
fT are known,
where C9 5 7.06 and C10 5 8.82. If FR and F
then S X j0 can be directly obtained by solving Eq. (16).
Consequently, F0 can be evaluated from Eq. (18),
yielding Fig. 2.
We follow Hansen and Rattray (1966) and use the
condition F0 5 0.1 to define the transition between wellmixed and partially mixed estuaries. To distinguish between partially mixed and highly stratified estuaries, we
use the condition F0 5 1.0 which corresponds to fresh
surface water extending to the mouth of the estuary. Our
classification scheme is obtained by plotting these transifT 5 0, the transition betional criteria on Fig. 2. When F
tween well-mixed and partially-mixed estuaries is
predicted to occur at FR 5 0.0018, and from partiallymixed to highly stratified at FR 5 0.113. The value of FR
fT increases, the infor both transitions increases as F
crease being more rapid for the transition from partially
mixed to highly stratified estuaries. These results are in
qualitative agreement with Fig. 2.7 of Geyer (2010).
5. Discussion
Together, Eqs. (16) and (18) provide new insight
into estuarine physics. Apart from broadly classifying
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GUHA AND LAWRENCE
parameters (MacCready 2007). For example, many estuaries are too sluggish to respond to fortnightly variability in tidal forcing.
To test the applicability of our classification scheme we
made use of the field data presented in Prandle (1985).
Using these data we have computed FR , FT , B/H, and
fT directly, and F0 from Eqs. (16) and (18); see Table 1.
F
We have compared the computed values of F0 with the
measured values in Fig. 3. The comparison is good confT can be desidering the accuracy to which FR and F
termined from field data.
fT are
It is interesting to note that if both FR and F
small then Eq. (16) reduces to S X j0 5 1 and Eq. (18)
reduces to
FIG. 2. Estuary classification diagram with lines representing
isocontours of F0 . The three regions are three types of estuaries:
(a) light gray is well mixed, (b) white is partially mixed, and (c) dark
gray is highly stratified or salt wedge. For data and expansion of
abbreviations, see Table 1.
estuaries into three categories, namely highly stratified,
partially mixed, and well mixed, these equations reveal
that under the given assumptions just two parameters,
fT , are needed to determine the stratification
FR and F
at the estuary mouth
pffiffiffiffiffiffiffiffiffiF
ffi 0 . The new nondimensional pafT 5 FT B/H reveals that the ‘‘tidal effect’’ is
rameter F
not simply represented by the tidal Froude number FT,
but the latter combined with the square root of the estuarine aspect ratio B/H. Moreover the equation set
fT . If the estuarine condition
predicts F0 , given FR and F
changes (for example, the river flow or the estuary
fT will change accorddepth), the parameters FR and F
ingly. These newly obtained Froude numbers will produce a new F0 , which reflects the response of estuarine
circulation and mixing to these changes. However, we
should be mindful of the fact that estuaries can be
‘‘sluggish’’ in their response to changes in the governing
F0 ’ 7FR2/3 .
(19)
The same result was obtained by MacCready and Geyer
(2010) by combining Knudsen’s relations (Knudsen
1900) with Eq. (9). Since Knudsen’s relations are derived from mass and salt balances and do not consider
momentum balance, Eq. (19) is an approximation of Eq.
(18). For a given FR , Eq. (19) always yields a higher
value for F0 than Eq. (18). The difference is small when
fT are small, but increases as FR and F
fT
both FR and F
increase (Fig. 4).
We also compare the theoretical results with the
field data of Prandle (1985) in Table 1 and Fig. 2.6
of Geyer (2010) in Fig. 4. We have chosen to plot
fT
fT 5 0 and 30, because most estuaries have F
Eq. (18) for F
within this range. Ideally, most of the partially and wellmixed estuaries should cluster within the gray region
fT 5 0 and 30, which is indeed the
bounded by the lines F
case. The most important aspect of this comparison is that
the theoretical curves follow the overall trend of the field
data. However, F0 tends to be overpredicted at high
values of FR , that is, mixing is underpredicted. This
discrepancy could arise if the values of F0 were measured at an upstream location, rather than at the mouth,
TABLE 1. Estimates of estuarine parameters calculated using the data of Prandle (1985) and values of B obtained from maps.
Equation (18) is used to obtain F0 (theory).
Estuary
Name
Abbreviation
FR
FT
B/H
fT
F
F0
F0 (theory)
Vellar
Columbia
James
Tees
Southampton Waterway
Tay
Narrows of the Mersey
Bristol Channel
V
C
J
Te
SW
Ta
NW
B
1.27
0.026
0.004
0.014
0.0012
0.014
0.0009
0.006
0.64
0.43
0.25
1.03
0.37
1.38
0.83
1.59
200
150
360
75
200
400
65
300
9.0
5.3
4.7
8.9
5.2
27
6.7
27
1.00
0.40
0.22
0.18
0.10
0.10
0.05
0.02
1.00
0.50
0.17
0.33
0.06
0.17
0.05
0.06
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FIG. 3. Comparison of stratification at the estuary mouth obtained
from theory or from field data.
or could be a result of increasing stratification inhibiting
vertical mixing.
A plot similar to Fig. 4 is presented in Geyer (2010,
Fig. 2.6). In this figure, a line [labeled Eq. (2.22)]
VOLUME 43
corresponding to F0 ’ 3FR2/3 provides a good fit to the
data. However, when Eq. (2.22) is evaluated using the
coefficients provided in Geyer (2010), the result
F0 5 8:73FR2/3 is obtained.
Finally, we refer to the assumptions behind our theoretical analyses and their consequences. We have
simplified the problem by assuming a tidally averaged
estuary with rectangular geometry. In real estuaries,
bathymetry can play a crucial role in determining estuarine circulation. Moreover, the appearance of just two
fT ) in our equations is a conseparameters (FR and F
quence of the empirical Eqs. (8) and (9). These equations also determine expressions for the coefficients
C1, C2, . . . , C10, given in Table 2. All these coefficients
depend upon the Schmidt number (i.e., Sc), which is an
empirical quantity that could conceivably vary from estuary to estuary, within a given estuary, or with time. Although Eqs. (8) and (9) are simple and elegant, they may
not be very realistic. In real estuaries, both KM and KS are
variables. Moreover, other empirical parameterizations
have shown that KM depends upon the Richardson number (MacCready and Geyer 2010). While the inclusion
of the Richardson number, or any other relevant parameter, might improve the predictability of the classification
scheme, the value of this improvement would have to be
weighed against the added complexity of the resulting
classification scheme.
FIG. 4. Comparison of the estuary classification scheme with the approximation F0 5 7FR2/3
in Eq. (19). The gray area indicates the region where estuaries should ideally cluster. Field
data from Geyer (2010) and Prandle (1985) are plotted for comparison with the theoretical
predictions.
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GUHA AND LAWRENCE
TABLE 2. List of coefficients used in different equations.
Coefficient
Value
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
1/3
1.67Sc
0.792Sc2/3
41.7a0a1CDSc21/3
0.868Sc21/3
36.2Sc2/3
41.7Sc1/3
4.08Sc1/3
3.57Sc2/3
5.43Sc1/3
5.21Sc2/3
6. Conclusions
The equations governing the physics of estuarine circulation have been presented in nondimensional form.
The two resulting nondimensional parameters are the
estuarine Froude number FR and the modified tidal
fT . Given these parameters, the nonFroude number F
dimensional salinity gradient at the estuary mouth S X j0
and the nondimensional salinity stratification (also at
the estuary mouth) F0 can be computed. The latter result forms the basis of a classification scheme that can be
used to predict whether an estuary is fully or partially
mixed, or highly stratified. The predictions of this classification scheme compare well with estuarine data.
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