Tidally averaged circulation in Puget Sound sub-basins

Estuarine, Coastal and Shelf Science 93 (2011) 305e319
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Estuarine, Coastal and Shelf Science
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Tidally averaged circulation in Puget Sound sub-basins: Comparison of historical
data, analytical model, and numerical model
Tarang Khangaonkar a, *, Zhaoqing Yang a, Taeyun Kim a, Mindy Roberts b
a
b
Pacific Northwest National Laboratory, Marine Sciences Division, 1100 Dexter Avenue North, Suite 400, Seattle, WA 98109, USA
Washington State Department of Ecology, PO Box 47600, Olympia, WA 98504-7600, USA
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 25 September 2010
Accepted 25 April 2011
Available online 10 May 2011
Through extensive field data collection and analysis efforts conducted since the 1950s, researchers have
established an understanding of the characteristic features of circulation in Puget Sound. The pattern
ranges from the classic fjordal behavior in some basins, with shallow brackish outflow and compensating
inflow immediately below, to the typical two-layer flow observed in many partially mixed estuaries with
saline inflow at depth. An attempt at reproducing this behavior by fitting an analytical formulation to
past data is presented, followed by the application of a three-dimensional circulation and transport
numerical model. The analytical treatment helped identify key physical processes and parameters, but
quickly reconfirmed that response is complex and would require site-specific parameterization to
include effects of sills and interconnected basins. The numerical model of Puget Sound, developed using
unstructured-grid finite volume method, allowed resolution of the sub-basin geometric features,
including presence of major islands, and site-specific strong advective vertical mixing created by
bathymetry and multiple sills. The model was calibrated using available recent short-term oceanographic
time series data sets from different parts of the Puget Sound basin. The results are compared against 1)
recent velocity and salinity data collected in Puget Sound from 2006 and 2) a composite data set from
previously analyzed historical records, mostly from the 1970s. The results highlight the ability of the
model to reproduce velocity and salinity profile characteristics, their variations among Puget Sound subbasins, and tidally averaged circulation. Sensitivity of residual circulation to variations in freshwater
inflow and resulting salinity gradient in fjordal sub-basins of Puget Sound is examined.
Ó 2011 Elsevier Ltd. All rights reserved.
Keywords:
modeling
fjords
partially mixed estuaries
analytical solution
3-D hydrodynamic model
unstructured grid
Puget Sound
FVCOM
1. Introduction
Puget Sound, the Strait of Juan de Fuca, and Georgia Strait,
recently defined as the Salish Sea, compose a large and complex
estuarine system in the Pacific Northwest portion of the United
States (U.S.) and adjacent Canadian waters. Pacific tides propagate
from the west into the system via the Strait of Juan de Fuca
around the San Juan Islands, north into Canadian waters through
the Georgia Strait. Propagation of tides into Puget Sound occurs
primarily through Admiralty Inlet (see Fig. 1(a)). This is a glacially
carved fjordal estuarine system with many narrow long and
relatively deep interconnected basins. The freshwater discharged
by 19 gaged rivers including those in Puget Sound, the inflows to
the Strait of Juan de Fuca, the freshwater discharge from the
* Corresponding author. Pacific Northwest National Laboratory, Marine Sciences
Division, 1100 Dexter Avenue North, Suite 400, Seattle, WA 98109, USA.
E-mail address: [email protected] (T. Khangaonkar).
0272-7714/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ecss.2011.04.016
Fraser River in Canadian waters, and substantial regional runoff
help create stratified two-layer conditions where the tidally
averaged circulation consists of outflow mixed brackish water in
the surface layer and inflow of saline water through the lower
layers.
Much of our present understanding of tidally averaged circulation in Puget Sound is based on analysis and interpretation of
considerable data collected since the 1950s and insights gained
from the application of a physical scale model of Puget Sound at the
University of Washington (Rattray and Lincoln, 1955). The historical
records of moored current meter and salinity profile observations
are extensive and date back to 1930. Cox et al. (1981) tabulated
known current observations, including periods of intensive monitoring from 1951 to 1956 and in the 1970s and 1980s. Ebbesmeyer
et al. (1984) and Cox et al. (1984) provided a synthesis and interpretation of these current measurements in Puget Sound. Using this
information, Ebbesmeyer and Barnes (1980) developed a conceptual model of Puget Sound which describes circulation in the main
basin of Puget Sound as that in a fjord with deep sills (landward sill
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T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
Fig. 1. (a) Oceanographic regions of Puget Sound and the Northwest Straits (Salish sea) including the inner sub-basinsdHood Canal, Whidbey Basin, Central Basin, and South Sound.
(b) Intermediate scale FVCOM grid along with composite current and salinity profile stations from Cokelet et al. (1990).
zone at Tacoma Narrows and a seaward sill zone at Admiralty Inlet)
defining a large basin, outflow through the surface layers, and
inflow at depth. Analysis of long-term current meter records by
Cannon (1983) showed a departure from the classic fjordal signatures seen in Hood Canal and Saratoga Passage to one where depth
of zero flow (where tidally averaged velocity crosses zero between
outgoing surface layer and inflowing deeper layer) was deeper in
the water column in the main basindabout 25% of depth, lower
than conventional fjords (10e15%) in the main basin. The depth of
peak inflow varied from near 50e75% of depthddeeper than classic
fjords, but higher in the water column than typical shallow estuaries. This behavior may be recognized as a transition between
a fjord and partially mixed estuary and is a characteristic feature of
Puget Sound circulation.
Nutrient pollution is considered a potential threat to the ecological health of Puget Sound. There is considerable interest in understanding the hydrodynamics and the effect of nutrient loads entering
Puget Sound and, given climate change and sea-level rise possibilities, how this balance may be altered in the future. This concern is not
new. Recognizing that pollutant build-up problems and climatic
changes are longstanding and require long-term application of
models, Cokelet et al. (1990) developed a numerical model consisting
of a branched system of two-layered reaches separated by exchange
zones to calculate refluxing, salinity concentrations time series, and
annual volume transports in nine sub-basins of Puget Sound. The
technique used estimates of annual freshwater runoff and composite
profiles of long-term mean currents and salinity profiles from historic
measurements in nine reaches to specify conservative mass transports for the Strait of Juan de Fuca and Puget Sound domain. A similar
branched system, consisting of seven, two-layered boxes (Box Model
of Puget Sound) was developed by Babson et al. (2006) based on the
work by Li et al. (1999). The basin characteristics, such as depth of
zero flow to determine the thickness of boxes representing specific
basins, were adopted from the composite information of Puget Sound
developed by Cokelet et al. (1990). A key simplification in these and
prior Puget Sound box models, including those by Friebertshauser
and Duxbury (1972) and Hamilton et al. (1985), is the fixed and
constant surface and bottom layer depth in each basin. Only a limited
number of hydrodynamic modeling studies exist that cover the entire
Puget Sound, including a laterally averaged, vertical two-dimensional
model used to study density intrusion into Puget Sound (Lavelle et al.,
1991), and a three-dimensional (3-D), structured grid model developed using Princeton Ocean Model (POM) code to study the variability of currents with a focus on complexities of the triple junction
site at the confluence of Admiralty Inlet, Possession Sound and the
Main (or Central) Basin of Puget Sound (Nairn and Kawase, 2002).
Water quality in Puget Sound, as indicated by conventional
parameters such as dissolved oxygen, nutrients (nitrate þ nitriteenitrogen (NO3 þ NO2) and phosphateephosphorus (PO4)), algae, and
fecal coliform bacteria, is generally considered to be good. However,
there are several specific locations where water quality appears
reduced due to low dissolved oxygen and fecal coliform bacteria
contamination. The areas with lowest dissolved oxygen levels include
southern Hood Canal, Budd Inlet, Penn Cove, Commencement Bay,
Elliott Bay, Possession Sound, Saratoga Passage, and Sinclair Inlet.
Historical observations of primary production suggest that phytoplankton growth in Puget Sound is closely coupled to the circulation
characteristics. This was demonstrated through the early work by
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
Winter et al. (1975). Subsequent studies and review of historic data
show that although spring and summer blooms occur regularly, the
potential for eutrophication impacts in the main basin is mitigated by
the presence of strong residual circulation and water renewal from
freshwater discharges and inflow of water from the Pacific Ocean.
However, poorly flushed inner basins and shallow embayments,
particularly in the southern end, show depleted surface nitrate
concentrations during the summer and very low oxygen concentrations at depth (Harrison et al., 1994; Newton and Van Voorhis, 2002).
Therefore to correctly simulate nutrient, algae and dissolved oxygen
balance for water quality management in Puget Sound, the ability to
reproduce observed characteristics of tidally averaged circulation
including bottom water renewal and surface water flushing is
essential.
In this paper, we first present an analytical approach to describe
tidally averaged current and salinity profiles in Puget Sound and
examine whether inter-basin variability can be captured through
variation in principle physical quantities, such as inflow, water
depth, and salinity gradient. We then present a numerical approach
based on an advanced 3-D baroclinic model of Puget Sound,
including the Salish Sea reaches in U.S. and Canadian waters, that
was previously developed (Khangaonkar and Yang, 2011; Yang and
Khangaonkar, 2010). A computationally efficient, intermediate
scale version of the model suitable for multi-year simulations was
adopted for this investigation and calibrated using data from 2006.
Quantitative model skill analysis is also presented in the form of
error statistics at available representative stations around Puget
Sound. A year-long application of the model was then used to
generate tidally averaged current and salinity distributions and
compared to composite current and salinity profiles developed by
Cokelet et al. (1990) from historical data. The comparison demonstrates the ability of the model to reproduce tidally averaged
circulation patterns and the ability to simulate inter-basin variability of mean currents and salinity profiles. The model was also
used to evaluate the sensitivity of tidally averaged circulation, peak
inflow, outflow, and surface-layer thickness to inflow and salinity
gradient in one of the sub-basins (Saratoga Passage) of Puget
Sound. Saratoga Passage was selected for this analysis. This basin is
influenced by Skagit River discharge which is the largest freshwater
discharge into Puget Sound.
2. Analytical formulation
Prior analytical efforts to describe Puget Sound’s mean circulation using an analytical approach were conducted nearly four
decades ago by Winter (1973) and Winter et al. (1975). They applied
Rattray’s (1967) general fjord circulation theory to the main basin of
Puget Sound and found approximate agreement in current
magnitudes but shallower depth of zero flow. To obtain a solution
in a closed form, we have adopted an analytical approach where
a theoretical formulation previously used by Hansen and Rattray
(1965), Dyer (1973), and MacCready (2004) for partially mixed
estuaries based on a constant eddy viscosity assumption was
expanded to incorporate an exponential variation with depth
similar to the form used by Rattray (1967).
Governing equations applicable to narrow, long fjordal estuaries
are the continuity and Reynolds momentum equations in a vertical
two-dimensional (xez) coordinate system, an equation of state
relating salinity to density, and the hydrostatic assumption for
pressure. Under steady-state conditions employing the Boussinesq
approximation, the simplified governing equations with the origin
located near the mouth of the estuary, the x direction pointed
toward the open ocean boundary, and the z axis being positive
upwards may be expressed as follows:
307
vu vw
þ
¼ 0;
vx vz
(1)
vu
vu
1 vP
v
vu
Km
;
u þw
¼ þ
r vx vz
vx
vz
vz
(2)
P ¼ r$g$ðh zÞ:
(3)
Equations (1) and (2) are continuity and momentum equations in
the x direction, assuming that the longitudinal momentum diffusion terms are small for long, narrow estuaries. Equation (3) is the
hydrostatic pressure, where u(x,z) and w(x,z) are velocities (m/s) in
the x and z directions, r is the density of water (kg/m3), h(x) is the
free-surface elevation (m), and Km(z) is the vertical eddy viscosity
coefficient (m2/s). For fjordal waters and partially mixed estuaries,
the equation of state may be approximated by:
r ¼ r0 $ð1 þ b$sÞ;
(4)
where s(x,z) is salinity in practical salinity units (ppt), and
b z 7.7 104/ppt.
The salt balance is given by the steady state advectionediffusion
balance:
vs
vs
v
vs
Ks
;
u þw ¼
vx
vz
vz
vz
(5)
where Ks(z) is the vertical eddy diffusion coefficient (m2/s).
Starting from the early work done by Pritchard (1952, 1954,
1956), variations of these basic equations have been used by
many researchers in the past to develop analytical solutions for
coastal plain estuaries, well-mixed estuaries, partially mixed
estuaries, and highly stratified fjordal estuaries (e.g., Hansen and
Rattray, 1965; Rattray, 1967; Jay and Smith, 1990a,b; MacCready,
2004).
We seek a formulation that would allow consideration of depthdependent Km(z) for application to fjordal conditions of the type
observed in Puget Sound with a varying degree of vertical shear and
mixing. Here, we introduce a non-dimensional depth variable
z ¼ z=H, where H is the water depth, and an exponential variation of
Km and Ks with depth, similar to the form used by Rattray (1967) is
defined by:
KmðzÞ ¼ Kmo$ea$z ;
(6)
KsðzÞ ¼ d$KmðzÞ ¼ d$Kmo$ea$z :
(7)
For narrow fjordal estuaries with a uniform cross section, the
solutions for u(z) and s(z) are obtained by solving governing
equations, assuming that the vertical velocity component w(x,z) is
small and the longitudinal momentum term is negligible relative to
the pressure gradient and the shear stress terms. The analytical
solution in closed form is proved below and the detailed derivation
is presented in Appendix A.
The tidally averaged longitudinal velocity solution, also referred
to here as the “variable Km solution,” is expressed by:
!
U az z2 2z 2
C1 eaz $ðaz þ 1Þ
þ C3;
uðzÞ ¼
$e
þ þ
$
Kmo
a a2 a3
Kmo
a2
where
vs
vx
U ¼ g$b$ $H3 ;
(8)
308
D¼
I1 ¼
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
C1 ¼
ea 1
ea $ð1 aÞ 1
2
; I3 ¼ ea $D 3 ;
; I2 ¼
a
a2
a
J1 ¼
1 2
2
2þ 3 ;
a a
a
I3 2$I2 2$I1
þ 2 þ 3
a
a
a
;
a$I2 þ I1
;
J2 ¼
a2
¼
U
u$Kmo $ðD$ea J1 Þ
2
;
ea $ð1 aÞ
J2 aa
U$D$ea
2$Kmo
C3
C1 ea ð1 aÞ
:
$
Kmo
a2
The variable Ks solution for the depth-varying component of
salinity is exhibited by:
Fig. 2. Comparison of predicted and observed velocity profiles at various stations in Puget Sound and the Straits (Data source: Cannon, 1983).
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
309
Table 1
Variation in input parameter valuesdanalytical model of Puget Sound.
Basin name
Physical properties
Admiralty Inlet
Shilshole Marina
Puget Sound main basin
Saratoga Passage
s0 ðzÞ ¼
Width,
W, (m)
River inflow,
Qr, (m3 s1)
Surface eddy viscosity
Kmo, (m2 s1)
Depth-penetration
factor, “a”
Salinity gradient, vs=vx, (ppt m1)
100
200
190
82
5446
5900
6177
4166
1.347
241
289
475
10 105
10 105
10 105
10 105
0.001
3.3
6.0
17.0
1.03 107
9.62 108
3.13 107
4 105
(
U
C1 e2az ð2az þ 5Þ
$
e2az
d$Kmo Kmo
2$Kmo
4a4
!
!
2
1 z
2z
2
2
þ 5 ðaz þ 2ÞÞ
þ
þ
a
a2 2a 4a2 8a3
)
eaz ðaz þ 1Þ
eaz
D1 þ D2;
ðC3 uÞ $
a
a2
a
where
M1 ¼
2a
e2a 1
e $ð1 2aÞ 1
; M2 ¼
;
2a
4a2
M3 ¼
D1 ¼
Analytical model parameters
Depth,
H (m)
a
e2a
d$Kmo
1
2a
4a2 2 þ 8a2 3
2
;
8a3
3$U
2$C1
; and
þ
Kmo,a4 Kmo$a3
D2 ¼
(9)
a
U
C1 ð2aM2 þ 5M1 Þ
þ
$
$
2$Kmo
Kmo
4a4
1 M3 2M2 2M1
2
þ
þ
ðaM
þ
2M
Þ
þ
2
1
a5
4a2
8a3
a2 2a
ðaI þ I Þ
I
þ 1 D1:
þðC3 uÞ $ 2 2 1
a
a
d$Kmo
Salinity profiles are then computed using the definition:
sðzÞ ¼ s0 ðzÞ þ s (where s0 ðzÞ and s are the depth-dependent and
depth-averaged components of s and the assumption that at the
seabed ðz ¼ 1Þ, salinity may be assumed equal to ocean salinity
(i.e., sð1Þ ¼ socn ).
Comparisons of measured currents in various parts of Puget
Sound using varying magnitudes of the exponential decay coefficient “a” and Kmo values are shown in Fig. 2 and listed in Table 1.
For calibration of the tidally averaged analytical model in Puget
Sound, we have used previously published data collected from
different parts of Puget Sound as part of an effort to characterize
temporal and spatial variability in circulation and large-scale
dynamics of the estuarine system (Geyer and Cannon, 1982;
Cannon, 1983). We selected four stations that highlight the
Fig. 3. (a) Tidal elevations at open boundaries: entrance of the Strait of Juan de Fuca (Black) and North End of the Strait of Georgia. (b) Basin-wide river inflowsdPuget Sound, the
Strait of Juan de Fuca, and Georgia Strait.
310
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
different characteristics within Puget Sound (Fig. 1(a)). The velocity
profile in Admiralty Inlet under the influence of mixing over the
entrance sill shows characteristics seen in coastal plain or
a partially mixed estuary that is generally well described using
a near-constant description of the eddy viscosity coefficient, Kmo.
As shown in Fig. 2, in Admiralty Inlet, the inflow at the bottom
peaks approximately at three-fourths (3/4 ) of the water depth level.
Further south in Puget Sound, the Shilshole and the Main Basin
sites show that the shape of the residual current profile has
changed. For the period of record, the peak inflow and zero flow
(crossover) occurs at a much shallower depths in the Main Basin
and Shilshole sites relative to Admiralty Inlet. Fig. 2 shows that the
variable Km model works well over most of the region. In the
relatively exposed Puget Sound Main Basin location, it is likely that
Fig. 4. (a) Example of comparison between simulated and observed water surface elevation at Seattle station. (b) Example of comparison between simulated and observed currents
in Skagit Bay in the surface middle and bottom layers of the water column.
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
wind mixing caused a reduction of the surface-layer outflow.
Hence, the predicted solution is found to overestimate the outflow
currents. In the sheltered region of the Saratoga Passage, the
surface layer appears to be intact, and the overall behavior, which is
like a classic fjord, is matched well over the entire water column.
The fit was achieved in two steps. After specifying physical properties of the basin as input (H, W, and Qr) and selection of a reasonable
but arbitrary value of surface eddy viscosity (Kmo ¼ 10 105), the
depth-penetration factor “a” was adjusted so peak inflow depth
matched data. The depth-averaged salinity gradient ðvs=vxÞ was then
adjusted until a best match to velocity magnitudes was obtained. For
a fixed “a” value and all other physical parameters remaining
unchanged, the ratio of vs=vx to Kmo controls exchange flow and
varies basin to basin, but is a constant in each basin.
311
This exercise was valuable in showing it is possible to aggregate
the effects of many dominant physical Puget Sound processes into
a few parameters. The development of characteristic surface outflow
and shallow inflow in the upper half of the water column could be
created through the analytical formulation of Km, which increased
exponentially with depth. The depth of peak inflow and layer depth
(or depth of zero flow) was controlled by the specified magnitude of
“a,” and peak outflow or inflow was shown to be sensitive to
components of exchange flow velocity UE (see Appendix A), which
includes depth H, Kmo, and the depth-averaged salinity gradient
vs=vx. If we assume that surface eddy viscosity (Kmo) within Puget
Sound may be set relatively uniform, the variation of circulation in
sub-basins is controlled by 1) hydraulic features, such as sills and
basin geometry, and 2) freshwater inflow. The combined effects
Fig. 5. (a) Simulated (line) and observed (circle) salinity profiles in February high-flow condition. (b) Salinity time series comparisons at Saratoga Passage, East Passage, and Hood
Canal stations, respectively (Julian Day 1 ¼ January 1, 2006).
312
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
Table 2
Model calibration error statistics.
a) For water surface elevation (2006)
Station
MAE (m)
RMSE (m)
RME (%)
Correlation (R)
Port Angeles
Friday Harbor
Cherry Point
Port Townsend
Seattle
Tacoma
0.22
0.24
0.30
0.22
0.26
0.27
0.27
0.30
0.36
0.28
0.31
0.33
7.5
8.0
8.6
7.0
7.2
7.6
0.944
0.950
0.953
0.965
0.976
0.977
Mean
0.25
0.31
7.65
0.961
b) For velocity
Station
MAE (m/s)
RMSE (m/s)
Correlation (R)
Surface
Middle
Bottom
Surface
Middle
Bottom
Surface
Middle
Bottom
Pickering Passage
Dana Passage
Swinomish Channel
Skagit Bay
0.16
0.26
0.26
0.30
0.11
0.26
0.33
0.21
0.10
0.26
0.30
0.18
0.20
0.30
0.32
0.38
0.14
0.30
0.37
0.29
0.13
0.31
0.34
0.22
0.686
0.919
0.754
0.652
0.914
0.943
0.555
0.798
0.831
0.913
0.466
0.795
Mean
0.23
0.28
0.769
c) For salinity
Station
MAE (ppt)
RMSE (ppt)
Admiralty Inlet Entrance (ADM2)
Admiralty Inlet North (ADM1)
Admiralty Inlet South (ADM3)
Puget Sound main basin (PSB)
East Passage (EAP)
Gordon Point/Tacoma Narrows (GOR1)
Hood Canal (HCB003)
Saratoga Passage (SAR003)
Nisqually Reach (NSQ)
Dana Passage (DNA)
0.60
1.04
0.81
1.29
0.66
0.67
0.72
0.63
0.79
0.96
0.85
1.23
0.97
1.86
0.98
0.93
0.93
1.10
1.00
1.15
Mean
0.82
1.10
MAE ¼ mean absolute error; RMSE ¼ root mean square error.
of which are included in specified values of “a” and the resulting
vs=vx, respectively.
It may be reasonable to assume that surface eddy coefficient
Kmo, ratio of eddy viscosity to diffusivity d, and depth-penetration
factor “a” are likely governed by intrinsic geometric and hydraulic
features of each basin for normal weather and wind conditions.
Once determined for each basin through parameter calibration or
site-specific measurements, they may be expected to remain relatively constant. Salinity gradient, however, will vary seasonally and
change with alterations to runoff and precipitation regimes with
potential changes in climate. In the current analytical formulation,
we have not attempted to relate vs=vx to inflow Qr. Therefore,
a direct application of the analytical solution to predict a response
of current and salinity profile to changes in freshwater inflow is not
possible. Without prior knowledge of these parameters, the exercise at this point may be viewed as a sophisticated fitting of solution
to the observed data in individual basins. As more data becomes
available, there is potential for both practical applications and
developing better understanding of circulation in fjord-like estuaries using this type of approach.
In the absence of direct measurements, we turn to a 3-D numerical
model with the ability to estimate eddy viscosity and eddy diffusivity
coefficients internally through turbulence closure schemes and to
compute longitudinal and vertical salinity gradients in baroclinic
mode for the entire domain. The reproduction of observed current
and salinity profiles (shape and magnitude) in Puget Sound using
a simple analytical formulation demonstrated here sets the standard
and performance expectations for the numerical model. The
numerical results may then be assimilated back into the analytical
model for future development and application.
3. Puget Soundd3-D circulation and transport numerical
model
3.1. Model setup
A multiscale model of Puget Sound with a grid size capable of
resolving small channels near river mouths to coastal open waters
was developed previously and has been applied on multiple
projects in connection with nearshore restoration actions for
improving the water quality and ecological health of Puget Sound
(Yang and Khangaonkar, 2010; Khangaonkar and Yang, 2011). The
Puget Sound model uses the finite volume coastal ocean model
Table 3
Compilation historical Puget Sound data (Cokelet et al., 1990).
Reach
Pillar Point
New Dungeness
Point Jefferson
East Passage
Colvos Passage
Gordon Point
Devil’s Head
Tala Point
Hazel Point
Saratoga Passage
Years of observation
Velocity
Salinity
1975e1978
1964, 1978
1972e1973,
1975e1978
1943, 1977,
1982e1983
1947, 1977
1945, 1977e1978
1945, 1978
1942, 1952,
1963, 1977e1978
1942, 1978
1943, 1970, 1977
Primary salinity data was
obtained from the University
of Washington’s field program
(1951e1956)
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
(FVCOM) developed at the University of Massachusetts (Chen et al.,
2003). FVCOM is a 3-D hydrodynamic model that can simulate
tidally and density-driven, and meteorological forcing-induced
circulation in an unstructured, finite element framework. The
unstructured-grid model framework of FVCOM is well suited to
accommodate complex shoreline geometry, waterways, and islands
in Puget Sound. FVCOM solves the 3-D momentum, continuity,
313
temperature, salinity, and density equations in an integral form.
The model employs the Smagorinsky scheme for horizontal mixing
(Smagorinsky, 1963) and the Mellor-Yamada level 2.5 turbulent
closure scheme for vertical mixing (Mellor and Yamada, 1982). The
model has been successfully applied to simulate hydrodynamics
and transport processes in many estuaries, coastal waters, and open
oceans (Zheng et al., 2003; Chen and Rawson, 2005; Isobe and
Fig. 6. (a) Comparison of simulated Year 2006 average velocity profiles at various Puget Sound sub-basins with composite data from Cokelet et al. (1990). (b) Comparison of
simulated Year 2006 average salinity profiles at various Puget Sound sub-basins with composite data from Cokelet et al. (1990).
314
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
Beardsley, 2006; Weisberg and Zheng, 2006; Zhao et al., 2006; Aoki
and Isobe, 2007; Chen et al., 2008; Yang and Khangaonkar, 2009;
Yang et al., 2010a,b; Khangaonkar and Yang, 2011)
For this study, we developed an intermediate (coarser) scale
version of the same domain with grid size varying from 350 m in
estuaries and bays to as large as 3000 m in open coastal waters (see
Fig. 1(b)). The grid extends north into Canadian waters up to
northern Strait of Georgia (south of the entrance to Johnstone Strait),
and west to Neah Bay at the entrance to the Strait of Juan de Fuca. A
sigma-stretched coordinate system was used in the vertical plane
with 10 terrain-following sigma layers distributed using a power law
function with exponent P_Sigma ¼ 1.5 with higher resolution near
the surface. This scale allows sufficient resolution of the various
major river estuaries and sub-basins while allowing year-long
simulations within 1e2 days of run times on a multiprocessor
cluster computer. The bathymetry was from a combined data set
consisting of data from Puget Sound digital elevation model
(PSDEM) (Finlayson, 2005) and data provided by the Department of
Fisheries and Oceans Canada covering the Strait of Georgia. The
model bathymetry within the Puget Sound portion of the model
south of Admiralty Inlet was smoothed to minimize hydrostatic
inconsistency associated with the use of the sigma coordinate
system with steep bathymetric gradients. The associated slopelimiting ratio dH/H ¼ 0.2 was specified within each grid element
inside the Puget Sound region following guidance provided by
Mellor et al. (1994) and using site-specific experience from Foreman
et al. (2009), where H is the local depth at a node and dH is change in
depth to the nearest neighbor.
The model setup was conducted using Year 2006 as the basis
since it was the most data-rich year for salinity, temperature, and
water quality information from Puget Sound. Tidal elevations were
specified along the open boundaries using XTide (harmonic tide
clock and tide predictor) predictions (Flater, 1996) at the Tatoosh
Island station located at the entrance of the Strait of Juan de Fuca
and the Campbell River station at the mouth of Johnstone Strait.
Fig. 3(a) shows tidal elevations forcing input specified at the open
boundaries. In this study, temperature and salinity profiles along
the open boundaries were estimated based on monthly observations conducted by the Department of Fisheries and Oceans Canada
(near the open boundaries). In the entire model domain, initial
temperature and salinity conditions were specified uniformly as
9 C and 31.5 ppt, respectively, and water surface and velocities
were set to zero. To obtain the final initial condition for the Year
2006 model run, the model was spun up with Year 2005 forcing
inputs. Sensitivity tests using constant tidal forcing and steady
freshwater flows showed that dynamic steady state over most of
the model domain with respect to velocity and salinity profile was
achieved in 6e7 months of simulation confirming that 1-year-long
simulation would be an adequate spin up period.
The model includes 19 rivers that are incorporated with the
resolution of estuarine distributary reaches. The Puget Sound
region experienced a significant flood event in November 2006,
which is reflected in the river discharge time series. In contrast, the
Fraser River inflow on the Canadian side of the domain, which is
significantly higher than the rest of the inflows into Puget Sound
and the Straits, shows a very different seasonal distribution patterndhigh flow in the late spring and summer and low flow in the
fall and winter. Fig. 3(b) shows a plot of basin-wide freshwater
discharges grouped by their discharge basins. The Whidbey Basin
consists of the three largest rivers, Skagit River, Snohomish River,
and Stillaguamish River, in Puget Sound and accounts for almost
70% of the total freshwater flow into Puget Sound.
Meteorological parameters for calculation of net heat flux in
FVCOM include: 1) downward and upward shortwave radiation,
2) downward and upward longwave radiation, 3) latent heat
flux, and 4) sensible heat flux. These meteorological parameters
were obtained from North American Regional Reanalysis (NARR)
data generated by the National Oceanic and Atmospheric Administration (NOAA) National Center for Environmental Prediction
(NCEP) and used to compute and specify net heat flux at the
surface. Wind stress within FVCOM is calculated based on the wellknown Large and Pond method (1981). In general, winds are mostly
southerly within Puget Sound with low speeds during summer
(<5 m/s) and high during winter (as high as 15 m/s). Winds can
reach gale-force easterly speeds, 17e24 m/s or higher, in the Strait
of Juan de Fuca (Holbrook et al., 1980). Local winds are known to be
affected by topography and accurate simulation of wind effects
would require specification of wind field at 1e5 km resolution. In
this application, representative wind field was specified using
NARR prediction from a station near main basin of Puget Sound.
3.2. Model calibrationd2006 data
The primary calibration effort was associated with refining and
smoothing of bathymetry; averaging of specified boundary forcing
salinity and temperature profiles; and adjustment of bed friction
until a stable model operation and best fit of predicted water surface
elevation (WSE), velocity, salinity, and temperature to observed data
at selected stations in Puget Sound was achieved. There are six realtime tidal stations maintained by NOAA as part their Physical
Oceanographic Real-Time Systems (PORTS) program throughout the
Straits and Puget Sound. Velocity data are quite limited in Puget
Sound. Acoustic Doppler Current Profiler (ADCP) data in South Puget
Sound, Skagit Bay, and Swinomish Channel were used for model
calibration (Yang and Khangaonkar, 2009). Monthly salinity and
temperature profiles collected by the Washington State Department
Fig. 7. (a) Tidally averaged salinity gradient ðvs=vxÞ time history at Saratoga Passage
station simulated by the numerical model of Puget Sound for Year 2006. (b) Tidally
averaged velocity profiles at Saratoga Passage station predicted using the analytical
model for salinity gradient ðvs=vxÞ varying from 0.5 105 ppt/m to 7 105 ppt/m as
provided by the numerical model. (c) Tidally averaged (monthly) velocity profiles
extracted from the numerical model directly for the Year 2006 simulation.
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
of Ecology as part of their ambient monitoring program throughout
Puget Sound were also used.
Examples of time series comparisons for WSE, velocity, and
salinity at selected stations are shown in Figs. 4(a), (b), and 5(b),
respectively, to illustrate the quality of match in phase and
magnitude. Error statistics at all stations analyzed are provided in
Table 2aec and provide a quantitative assessment of the model’s
ability to reproduce observed oceanographic parameters.
As shown in Table 2a, mean absolute errors (MAEs) and root mean
square errors (RMSEs) for WSE of all the stations are 0.25 m and
0.31 m, respectively. Relative mean errors (RMEs), defined as the ratio
of MAE to the mean of daily tidal ranges, were within 10%. Sensitivity
tests indicate that the main source of error is likely the error in
specified boundary elevations at the Georgia Strait boundary.
Velocity data available for model calibration in 2006 were
limited. The ADCP data were collected within narrow, long channels
with dominant longitudinal characteristics. For simplicity, comparisons were made between the model results and observed data
along the major axis of tidal currents at the surface, middle, and
bottom layers of the water column as shown in Fig. 4(b). The largest
errors are in the surface layer, and smallest errors are in the bottom
layer. Larger errors in the surface layer are likely due to limited
accuracy associated with the use of wind field specified using NARR
prediction from a representative station near main basin of Puget
Sound. The overall MAE and RMSE for all four stations are 0.23 m/s
and 0.28 m/s, respectively, as shown in Table 2b.
Salinity and temperature predictions inside Puget Sound are
sensitive to specified boundary temperature and salinity profiles. A
review of the data from monthly profiles collected at the boundaries
showed that seasonal salinity and temperature variations were only
notable in the upper 50 m of the water column and nearly constant
below throughout the year. Open boundary temperature and salinity
values were set constant at 7.4 C and 34 in the Strait of Juan de Fuca
and 9.3 C and 30.6 in Georgia Strait. Although year-long time
histories were not available, salinity and temperature profiles were
collected on a monthly basis at 25 monitoring stations within Puget
Sound. In this study, we selected 11 stations representing the subbasins in Puget Sound for temperature and salinity profile comparisons. Fig. 5(a) shows a comparison between simulated and observed
salinity profiles in February 2006 (high-flow condition). Table 2c
shows that MAE and RMSE for salinity profiles at all stations were
0.81 and 1.0 ppt respectively.
In addition to salinity profiles in Fig. 5(a), a comparison of predicted and observed salinity is presented in the form of time series
data from Saratoga Passage, Hood Canal, and East Passage subbasins in Fig. 5(b).
4. Tidally averaged velocity profilesdcomparison with
historical data
Capability of the Puget Sound hydrodynamic model of reproducing tidally averaged circulation is of importance as this drives the
mean transport and residence times, thereby influencing overall
water quality. This requires a comparison of low-passed, tidally
averaged currents and salinity with model results. The available
current meter records from Year 2006 were from inner sub-basins
not particularly well suited for evaluation of the tidally averaged
transport in key locations such as Admiralty Inlet, Saratoga Passage,
Main Basin, and Hood Canal. Therefore, we turned to historical
records of currents and salinity measurements. As mentioned in the
Introduction, much of our understanding of circulation in Puget
Sound is based on synthesis and interpretation of historical data
mostly from 1951 to 1956 and the 1970s and 1980s. Cokelet et al.
(1990) used this information to develop composite vertical current
and salinity profiles in each reach from short-term measurements.
315
Table 3 shows the years of observations at the respective stations,
and the station locations are indicated in Fig. 1(b).
A one-to-one comparison with model results corresponding to
Year 2006 and Cokelet et al. (1990) data is not appropriate as this
composite representation of Puget Sound circulation was constructed with data from multiple years. However, the ability to
reproduce the characteristic features using the Puget Sound
Circulation and Transport Model is encouraging. Fig. 6(a) shows
a comparison of velocity profiles from 10 stations in Puget Sound
and the Strait of Juan de Fuca digitized from Cokelet et al. (1990)
and the model results from Year 2006 averaged at the respective
station location for the entire year of simulation. The velocity data
collected over multiple years was not averaged at each depth and
hence is shown as many discrete points. Fig. 6(b) shows a comparison of model results to salinity data also collected over multiple
years but plotted after averaging at each depth.
The historical composite velocity profiles show spatial variation
and differences that also are recognizable in the simulated results.
As postulated previously and based on observed data, Saratoga
Passage in Whidbey Basin and Hazel Point in Hood Canal exhibit
classical fjord behavior, and the model reproduces the shallow
brackish outflow layer with inflow high up in the water column.
The comparisons of velocity profiles at other stations show
appropriate changes in characteristics with a reasonable agreement
among the trends and magnitudes. For example, the tidally averaged currents are nearly unidirectional in Colvos Passage (north)
and East Passage (south), respectively. Despite limitations of setting
the model boundary at Neah Bay in the Strait of Juan de Fuca, the
velocity profile characteristics that resemble partially mixed estuaries with inflow of saline water from the coast, peaking at threefourths (3/4 ) depths, and outflow of combined Puget Sound and
Canadian freshwater discharges through the surface appear to be
correctly reproduced. As seen in Fig. 6(b), a salinity simulation
correctly converges to a stable solution of approximately 30e31
throughout Puget Sound. This also illustrates the ability of the 10layer intermediate scale model to develop stratified Puget Sound
conditions, including the sub-basins and inner passages.
5. Results and discussion
Despite mixing and recirculation induced by the sills and subbasins, the tidally averaged salinity and current profiles in Puget
Sound continue to retain many major fjord-like characteristics. As
seen in the numerical model comparison to historical data and
calibration using Year 2006 salinity profiles (Figs. 6 and 7),
a distinct stratified upper layer, varying between 5 and 20 m in
depth (z5e15% of the water depth), is maintained. Salinity variation below this surface layer is uniform with <1 ppt variation and
determined primarily by the salinity of near-bed inflow waters
from the Strait of Juan de Fuca near the mouth of Admiralty Inlet. In
pronounced fjordal sub-basins, such as Saratoga Passage and Hood
Canal, and inner reaches within south Puget Sound, freshwater
inflow relative to basin size can be high. In these sub-basins, during
periods of high runoff surface strong stratification with brackish
salinity of 20e30 is seen within the shallow upper layer. The ability
to simulate this response, as seen in the profiles and time series
comparisons, demonstrates the ability of the FVCOM with the
Mellor-Yamada level 2.5 turbulent closure scheme to simulate
fjordal estuaries such as Puget Sound.
Tidally averaged peak outflow velocities at the surface vary from
as <0.1 m/s in the inner basins, such as Hood Canal and parts of
South Puget Sound to 0.1e0.2 m/s in the Main and Whidbey Basins
to as high as z0.3 m/s in Strait of Juan de Fuca. Peak inflow
velocities in fjordal basins of Hood Canal and Saratoga Passage are
small (0.03e0.05 m/s), distributed over a larger depth relative to
316
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
outflow, and occur at relatively stable depths varying between 5%
(Saratoga Passage) and 10e30% (Hood Canal) of water depth based
on location relative to the sill. The depth of peak inflow drops to
near mid-depth (z50% water depth) in the Puget Sound main basin
at the Shilshole and Point Jefferson mooring locations. It is only in
regions with high energy mixing in the Strait of Juan de Fuca and
Admiralty Inlet where profiles show characteristics matching those
of partially mixed estuaries with inflow at z75% or greater depths
peaking at z0.20 m/s.
Using the analytical approach in combination with numerical
model results reconfirmed that river runoff is a key driver of the
mean circulation, and the mixing and stratification is dependent on
mean circulation as opposed to tidal currents. Lumping of important
physical estuarine processes into a few parameters allowed a closedform analytical solution, and, with the help of depth-varying eddy
coefficients, it was possible to fit observed data despite site-specific
complexity. The solution presented is limited by the fact that depthpenetration parameter “a” must be externally specified along with
the choice of Kmo. Yet, once the combination of Kmo and “a” is
determined as basin-specific parameters, the solution then is
primarily controlled by salinity gradient vs=vx, which, in turn, is most
directly controlled by freshwater inflow Qr. To illustrate this further,
daily average values of vs=vx at Saratoga Passage were extracted
from the Year 2006 model results and plotted as a function of Skagit
River inflow. The correlation between salinity gradient and river flow
is evident and is shown in Fig. 7(a). Fig. 7(b) shows that for the basinspecific values of eddy viscosity parameters (Kmo, “a”), the magnitude of the tidally averaged current profile is directly proportional to
the salinity gradient (Equation (8)). This analytical model result is
validated in Fig. 7(c), which shows monthly averaged velocity
profiles at the same location extracted from the numerical model
simulation. The salinity gradient varied from 0.5 106 to
7 106 ppt/m, and the simulated tidally averaged outflow velocity
at the surface varied from 0 to 0.2 m/s. The depth of zero flow in the
analytical model is a constant determined by the choice of Kmo and
“a”. The numerical model result shows that in sheltered locations
such as Whidbey Basin, which receives high amounts of freshwater,
the depth of upper layer may remain relatively stable (z8 m in
Saratoga Passage in 2006).
(z34 ppt) sets up the near-bed salinity of the water that enters
Puget Sound at Admiralty Inlet.
Acknowledgements
The development of this model of Puget Sound was partially
funded through a grant from the U.S. Environmental Protection
Agency (EPA) and Washington State Department of Ecology. Partial
funding also was received via a grant from the U.S. Department of
Energy as part of the Energy Efficiency and Renewable Energy
program. We would like to acknowledge Mr. Ben Cope of the EPA
and Ms. Karol Erickson for their encouragement and support. We
also would like to acknowledge comments provided by reviewers
from the model development technical advisory team with representatives from King County, the University of Washington, and
other stakeholders.
Appendix A. Analytical solution for mean circulation in fjordlike estuaries
For a narrow fjordal estuary with a uniform cross section, the
vertical velocity component w(x,z) is small, and the longitudinal
momentum term is negligible relative to the pressure gradient and
the shear stress terms (Dyer, 1973; MacCready, 2004) such that
Equation (2) reduces to:
0 ¼ 1 vP
v
vu
Km
:
þ
r vx vz
vz
Seeking an exact solution in an integral form, as opposed to a power
series approximation by Rattray (1967), we adopted the approach
used by MacCready (2004). In this approach, the velocity u(x,z) and
salinity s(x,z) are split into depth-averaged (u and s) and depthvarying (u0 and s) parts, respectively, which is expressed as:
uðx; zÞ ¼ uðxÞ þ u0 ðx; zÞ and sðx; zÞ ¼ sðxÞ þ s0 ðx; zÞ:
Using Equations (3) and (4) and assuming that
the pressure gradient may be written as:
Tidally averaged circulation and transport processes in Puget
Sound were analyzed using a combination of analytical and
numerical models. An analytical solution was derived using an
exponential form of the eddy viscosity coefficient and was shown
to fit observed historical data in Puget Sound. Sensitivity tests with
the analytical model highlighted the importance of freshwater
discharge to the surface waters of Puget Sound. Mixing and
development of stable stratified layers in Puget Sound is determined not by tidal currents, but by tidally averaged circulation and
flushing. This was confirmed by the 3-D hydrodynamic model of
Puget Sound, the Straits of Juan de Fuca, and Georgia Strait (Salish
Sea) developed using the unstructured-grid coastal ocean modeling
tool FVCOM. Composite salinity and velocity profiles from
a historical data set were used to validate the Puget Sound model’s
ability to reproduce the tidally averaged structure of Puget Sound
circulation. The model results showed that surface salinity in Puget
Sound responds rapidly to freshwater discharge, but salinity at
depth is almost entirely determined by the salinity and exchange in
the Strait of Juan de Fuca and Georgia Strait. The salinity gradient
between Canadian waters in Georgia Strait, which receives
considerable freshwater discharge (z30 ppt), and the saline Pacific
Ocean waters near the entrance to the Strait of Juan de Fuca
ðvs0 =vxÞ
1 vP
vh
vs
v
vu
KmðzÞ
:
¼ g þ g$b$ z ¼ r0 vx
vx
vx
vz
vz
6. Conclusion
(A1)
(A2)
ðvs=vxÞ,
(A3)
If Km were to be assumed constant (Kmo) as in the case of coastal
plain estuaries or even partially mixed estuaries, eliminating the
pressure gradient terms between equations (A1) and (A3) and
taking another derivative with z eliminates the x-dependent terms
and presents:
v3 u
g$b vs
¼
$
:
Kmo vx
vz3
(A4)
Similarly, if Ks were assumed constant (for simplicity, the eddy
diffusivity is assumed to be a ratio of vertical eddy viscosity
Ks ¼ d$Kmo) per Pritchard (1954), for partially mixed systems,
Equation (5) may be reduced to:
u0 $
vs
v 2 s0
zðd$KmoÞ 2 :
vx
vz
(A5)
The development above culminating in Equations (A4) and
(A5) is identical to that presented in MacCready (2004), where
a solution for (A4) and (A5) in the form of velocity and salinity
profiles through direct integration. Appropriate free-surface and
seabed boundary conditions are used, and there are net-flow
and salt-flux balance considerations across the selected cross
section.
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
We seek a formulation that would allow consideration of depthdependent Km(z) for application to fjordal conditions of the type
observed in Puget Sound with a varying degree of vertical shear and
mixing. Eliminating the vh=vx term from Equation (A3) and taking
the depth dependence of Km into consideration results in:
vs
v2 Km vu
vKm v2 u
v3 u
g$b$ ¼
$
þ
Km$
:
þ
2$
$
vx
vz vz2
vz2 vz
vz3
where a is a measure of the depth penetration of the circulation,
and d is the ratio of Ks/Km. Note that z varies from 0 to 1 and Ks
and Km increase with depth. Substituting Equation (A7) into (A6)
expresses:
U ¼ Km$
(A6)
Here, we introduce a non-dimensional depth variable z ¼ z=H,
where H is the water depth, and an exponential variation of Km
and Ks with depth similar to the form used by Rattray (1967)
defined by:
KmðzÞ ¼ Kmo$ea$z ;
(A7)
KsðzÞ ¼ d$KmðzÞ ¼ d$Kmo$ea$z ;
(A8)
317
vu
a2 $
v2 u
!
v3 u
þ 2$a$ 2 þ 3
vz
vz
vz
vs 3
and U ¼ g$b$ $H :
vx
(A9)
The solution for u(z) is obtained by integrating Equation (A9)
with respect to z subject to the conditions that vu=vz ¼ 0 at z ¼ 0
(free-surface zero shear assumption) and u(z) ¼ 0 at z ¼ 1 (zero
velocity at the seabed). Also, the depth-averaged velocity,
Z 0
Qr
¼
u ¼
uðzÞdz, where Qr is the freshwater river inflow
H$W
1
and H and W are the water depth and width of an equivalent
Fig. A1. (a,b) Along channel velocity “u” and profiles computed using variable Km. Solutions are plotted against non-dimensional depth z. (a) shows velocity profiles for a partially
mixed estuary using a small value of decay “a” (0.01) and typical values of Kmo ¼ 10 105. (b) shows velocity profile matching typical fjordal characteristics as in Rattray (1967)
using a higher magnitude of eddy viscosity and depth-penetration factor. (c, d) Corresponding salinity solutions.
318
T. Khangaonkar et al. / Estuarine, Coastal and Shelf Science 93 (2011) 305e319
estuary of rectangular cross section. The tide-averaged longitudinal
velocity solution, also referred to here as the “variable Km solution,” is expressed by:
C1 eaz $ðaz þ 1Þ
$
Kmo
a2
uðzÞ ¼
U
$eaz
Kmo
z
2
a
þ
2z 2
þ
a2 a3
!
þ C3;
(A10)
I1 ¼
1 2
2
2þ 3 ;
a a
a
ea 1
ea $ð1 aÞ 1
2
; I3 ¼ ea $D 3 ;
; I2 ¼
a
a2
a
J1 ¼
I3 2$I2 2$I1
þ 2 þ 3
a
a
a
C1 ¼
;
J2 ¼
0 v
vs
KsðzÞ
¼ a$u0 ðzÞ;
vz
vz
(A11)
where a ¼ H2 $ðvs=vxÞ, u0 ¼ ½uðzÞ u, and u(z) is taken from
Equation (A10). The integration of Equation (16) was completed
using the boundary conditions that there is no salt exchange across
the free-surface boundary, i.e., vs0 =vz ¼ 0 at z ¼ 0 and the depth
R0
average of s0 ¼ 0, or ð 1 s0 ðzÞdz ¼ 0Þ. The variable Ks solution for
the depth-varying component of salinity is formulated by:
where
D¼
MacCready (2004) in combination with the constant Km cubic
velocity profile to develop the quintic profile for salinity. Following
the treatment used by MacCready (2004), but with the exponentially varying Ks assumption of Equation (A8), the salt balance
formulation may be written as:
s ðzÞ ¼
U
C1 e2az ð2az þ 5Þ
$
e2az
2$Kmo
4a4
!
!
2
1 z
2z
2
2
z
þ
þ
ða
þ
2Þ
þ
a5
a2 2a 4a2 8a3
)
eaz ðaz þ 1Þ
eaz
D1 þ D2
ðC3 uÞ $
a
a2
d$Kmo Kmo
a$I2 þ I1
;
a2
U
u$Kmo $ðD$ea J1 Þ
2
;
ea $ð1 aÞ
J2 aa
U$D$ea
C
ea ð1 aÞ
C3 ¼
:
1 $
2$Kmo Kmo
a2
Note that in the Equation (A10) and subsequent developments
(shown below), it is understood that u, and s are functions of (x, z),
vs=vx and constants of integration with respect to z, (C1, C2, and C3)
are applicable only for a specific location (xo). In the above equations, the variation of Km in the vertical is governed by the value of
the depth-penetration factor, a.
Longitudinal velocity u(z) per equation (A10) is a function of
physical parametersddepth-averaged velocity u ¼ Qr =ðH$WÞ and
depth-averaged longitudinal salinity gradient vs=vx. The solution
also depends on model parameters eddy viscosity Km and depthpenetration coefficient “a”. For small magnitudes of a (jaj 0.01),
Km is nearly constant with respect to z, and the solution reduces to
that provided by MacCready (2004). Fig. A1(a) shows the analytical
solution of the longitudinal velocity per Equation (A10) using a small
value of a ¼ 0.01. A corresponding constant Km solution for
partially mixed estuaries is also plotted in terms of u and exchange
flow velocity UE ¼ ðg$b=KmÞðvs=vxÞðH 3 =48Þ as presented by
MacCready (2004, 2007). This is also recognized as the classic cubic
profile of Hansen and Rattray (1965). It is noted that exchange flow is
related to U in the variable Km formulation. For typical conditions
encountered in partially mixed and deep, narrow estuaries, the ratio
ðu=UE Þ 1 results in a fixed-profile shape such that inflowing
current always occurs at a constant depth, z ¼ 0.75. This limits the
use of the constant Km solution to typical fjords, where peak inflow
may occur much higher in the water column. Fig. A1(b) shows
a profile that matches the classic fjordal circulation description (e.g.,
Rattray, 1967), where the main circulation is restricted to a strong
outflow in the upper layer with an inflow immediately below the
pycnocline. This was obtained using a higher magnitude of depthpenetration factor in the variable Km solutiondEquation (A10).
Pritchard (1954, 1956) showed tidally averaged salt conservation for coastal and partially mixed estuaries is dominated by
the balance between advective shear and vertical diffusion. This
formulation was used by Hansen and Rattray (1965) and
(
a
0
(A12)
where
M1 ¼
2a
2a
e 1
e $ð1 2aÞ 1
;
; M2 ¼
2a
4a2
2
;
8a3
a
3$U
2$C1
D1 ¼
; and
þ
d$Kmo
Kmo$a4 Kmo$a3
a
U
C1 ð2aM2 þ 5M1 Þ
D2 ¼
þ
$
$
d$Kmo
2$Kmo
Kmo
4a4
e2a
M3 ¼
1
2
2
þ
2a 4a2 8a3
1 M3 2M2 2M1
2
þ
þ
ðaM
þ
2M
Þ
þ
2
1
a5
4a2
8a3
a2 2a
ðaI þI1 Þ
I
þ 1 D1:
þ ðC3 uÞ $ 2 2
a
a
(A13)
Salinity profiles are then computed using the definition:
sðzÞ ¼ s0 ðzÞ þ s and the assumption that, at the seabed ðz ¼ 1Þ,
salinity may be assumed equal to ocean salinity (i.e.,sð1Þ ¼ socn ).
Typical profiles of salinity for a partially mixed estuary are shown in
Fig. A1(c). For a small value of the depth-penetration factor
(a ¼ 0.01), the solution reduces to the classic quintic profile
(Hansen and Rattray, 1965; MacCready, 2004). Fig. A1(d) shows that
Equation (A13) may be used to generate the shallow brackish layer
at the surface matching the classic fjordal circulation description
(e.g., Rattray, 1967), using a higher magnitude of depth-penetration
factor in the variable Km solution.
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