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TRANSCRIPT
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PAGN:Uday SCAN:v4soft
0278-4343/$ - se
doi:10.1016/j.cs
�Correspondi503-748-1273.
E-mail addre1Now at Nat
Survey, Silver S
Continental Shelf Research ] (]]]]) ]]]–]]]
www.elsevier.com/locate/csr
OFA cross-scale model for 3D baroclinic circulation inestuary–plume–shelf systems: I. Formulation and skill
assessment
Yinglong Zhang, Antonio M. Baptista�, Edward P. Myers, III1
OGI School of Science & Engineering, Oregon Health & Science University, 20000 NW Walker Road, Beaverton, OR 97006, USA
O
ECTED PRAbstract
Challenges posed by the Columbia River estuary–plume–shelf system have led to the development of ELCIRC, a
model designed for the effective simulation of 3D baroclinic circulation across river-to-ocean scales. ELCIRC uses a
finite-volume/finite-difference Eulerian–Lagrangian algorithm to solve the shallow water equations, written to
realistically address a wide range of physical processes and of atmospheric, ocean and river forcings. The numerical
algorithm is volume conservative, stable and computationally efficient, and it naturally incorporates wetting and drying
of tidal flats. ELCIRC has been subject to systematic benchmarking, and applied to the description of the Columbia
River circulation. This paper motivates and describes the formulation, presents and critically analyzes the results of
selected benchmarks, and introduces ELCIRC as an open-source code available for community use and enhancement.
A companion paper describes the application of ELCIRC to the Columbia River.
r 2004 Elsevier Ltd. All rights reserved.
Keywords: Mathematical modeling; Ocean, coastal and estuarine circulation; Eulerian lagrangian methods; Finite volumes; Finite
differences; Semi-implicit methods
R 4951
53
COR1. Introduction
Tightly coupled estuary–plume–shelf systemspose cross-scale modeling challenges that are stillinsufficiently addressed by the communities devel-
UN55
57
e front matter r 2004 Elsevier Ltd. All rights reserve
r.2004.07.021
ng author. Tel.: +1-503-748-1147; fax: +1-
ss: [email protected] (A.M. Baptista).
ional Ocean Service, NOAA, Office of Coastal
pring, MD 20910-3282, USA.
oping open-source marine models such as POM(Blumberg and Mellor, 1987; Oey and Mellor,1993), ROMS (Haidvogel et al., 2000), SEOM(Iskandarani et al., 2003) and QUODDY (Lynchet al., 1996). To illustrate these challenges, weconsider briefly the Columbia River, a complexsystem that requires extensive, systematic model-ing due to its central role in the economy and wayof life of the Pacific Northwest of the UnitedStates.
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61d.
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With the second largest annual river dischargein the United States, the Columbia River is subjectto highly variable forcings from river, ocean andatmosphere, and is characterized by a richdiversity of circulation regimes and by strongphysical gradients. The freshwater plume is amajor regional oceanographic feature, controlledat various scales by coastal winds, freshwaterdischarges, tides and bathymetry. The estuary isshallow, except for two deep channels—the longestof which acts as the major conduit for freshwaterdischarge. Extensive wetting and drying (e.g., Fig.1a, b) occurs both in the estuary main stem and inecologically important lateral bays. In the chan-nels, velocities may reach 5m/s in ebb, and tidalellipses are topographically constrained. Hydraulicresidence times are short, and residual velocities
UNCORREC
(c)
(a)
Fig. 1. Among the many modeling challenges posed by the Columbia
(a. SAR image; b. ELCIRC salinity simulation, with white areas sh
estuarine and plume processes and features, including fronts (c. SAR
10,000 particles after 48 h of dispersal). SAR images are r Canadia
Comprehensive Large Array-data Stewardship System (CLASS).
OF
are strong and highly variable in space and time.Salinity intrusion is compressed, reaching at mostabout 48 km upstream of the mouth, while tidalinfluence is felt all the way to the first dam, 232 kmupstream. Strong, largely salinity-driven stratifica-tion fosters complex and highly variable barocliniccirculation patterns, and creates the opportunityfor estuarine turbidity maxima to develop in bothchannels of the main stem.The dynamics of the plume and the estuary are
inextricably linked, through active tidal exchangeand strong frontal structures extending across themouth (Fig. 1c, d). Shelf winds dramatically affectthe low-frequency water levels in the estuary.Through both upwelling and control of plumeorientation and attachment to the coast, shelf
TED PR 67
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(b)
(d)
River are the extensive wetting and drying of estuarine tidal flats
owing drying of tidal flats) and the tight interconnectivity of
image; d. ELCIRC simulations showing fronts through loci of
n Space Agency 2002, and are shown courtesy of the NOAA
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winds also impact estuarine budgets of heat andsalt.
Systematic, detailed numerical modeling of theColumbia River is being conducted in the contextof CORIE, a multi-purpose coastal margin ob-servatory (Baptista et al., 1998; Baptista et al.,1999; Baptista, 2002) that produces both dailyforecasts and multi-year hindcast simulations. Werecognized early on that setting a boundarycondition at or near the mouth of the ColumbiaRiver would be a daunting challenge (e.g., Fig. 1c,d), with unlikely odds of success—and thus wemade the decision to model the estuary and plume(thus, the shelf) as a whole. This was also mostconsistent with the scientific and managementissues that the CORIE modeling system isdesigned to address.
The broad context of the CORIE goals, thecomplexity of the Columbia River, and thedecision to address the Columbia River dynamicsacross estuarine–plume–shelf scales, combine tocreate a set of challenging modeling requirements,including:
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Covering geographically extensive domains (riv-er-to-ocean scales), while retaining high spatialresolutions in multiple localized regions.
�
77 CCovering multiple decades, at sub-hour resolu-tion.
�79
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REEnabling state-of-the-art 3D representations ofriverine, estuarine and ocean circulation pro-cesses, including advection-dominated flows,sharp density gradients, and wetting and drying.�
8385
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952Baroclinic terms and Mellor and Yamada (1982) turbulence
closure equations have recently been included in UnTRIM (V.
Casulli, private communication).
UNCORAchieving sufficient computational efficiency toenable both operational forecasting and crea-tion of long-term (multi-year) simulation data-bases, without sacrificing process representationor space–time resolution.
Early attempts by our group to use establishedmodels such as ADCIRC (Luettich et al., 1991),POM and QUODDY did not meet the aboverequirements (unpublished work); reasons varied,but commonly included cost inefficiencies due totime step constraints associated with the treatmentof advection. Circa (1999), an extensive review ofthe existing literature, revealed the existence of anunstructured-grid code (UnTrim, Casulli and
TED PROOF
Zanolli, 1998) the numerical strategy of which—and, in particular, the treatment of advection—appeared ideally suited for the task. However,neither UnTRIM nor its earlier, structured-gridversion (TRIM, Casulli and Cheng, 1992) wereavailable as free, open source software. UnTRIMalso did not include a baroclinic component2 andboth UnTRIM and TRIM were overly simplisticrelative to physical processes such as turbulence(e.g., direct specification of vertical mixing2) andair–water exchanges (e.g., no heat balance terms),which are essential in the Columbia River and inmany other marine modeling applications.We have since undertaken the development of
ELCIRC, inspired by the UnTRIM formulation,but independently coded and with expandedprocess representation. Like UnTRIM, ELCIRCsolves the shallow water equations using a semi-implicit Eulerian–Lagrangian finite volume/finitedifference method reliant on horizontally unstruc-tured grids and unstretched z-coordinates. How-ever, ELCIRC differs algorithmically fromUnTRIM in the treatment of tangential velocitiesand of transport quantities (salinity and tempera-ture). In addition, ELCIRC allows for the use ofstate-of-the-art turbulence closure schemes (Um-lauf and Burchard, 2003), includes terms for thetidal potential and atmospheric pressure gradients,and provides a detailed description of air–waterexchanges (see Appendix A).ELCIRC has been subject to extensive bench-
marking and has been applied to the description ofimportant Columbia River features and processes.A limited number of researchers have usedELCIRC in other applications (Robinson et al.,2004; Myers and Aikman, 2003; and Pinto et al.,2003). Based on the benchmarks and pilotapplications, our current assessment is that EL-CIRC is a functional code, generally well adjustedto the modeling requirements that we set forth,and with contrasting characteristics relative toexisting coastal circulation community models,especially in the treatment of advection andwetting-and-drying, and in the order of conver-
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gence of the numerical solution. As a natural nextstep, ELCIRC is being released as an open-sourcecode (CCALMR, 2003), with the expectation thatit will anchor a modeling framework that willevolve in robustness and disciplinary scope.
The present paper constitutes a comprehensivereference for version 5.01 of ELCIRC (henceforth,ELCIRC_v5.01), describing its formulation andcritically presenting the solution of several syn-thetic but demanding benchmarks. A companionpaper (Baptista et al., 2004) will describe in detailaspects of the simulation of the 3D barocliniccirculation in the Columbia River estuary andplume—by far the most extensive application ofELCIRC to date.
After this Introduction, Section 2 presents thephysical and mathematical formulation of themodel, with the air–water exchange formulationsummarized in Appendix A. The numericalstrategy is detailed in Section 3, and Section 4describes the performance of ELCIRC against arange of synthetic benchmark tests. Finally,Section 5 presents a road map for future develop-ments.
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UNCORREC2. Physical formulation
2.1. Governing equations
We solve for the free surface elevation, 3D watervelocity, salinity and temperature, using a set of sixhydrostatic equations based on the Boussinesqapproximation, which represent mass conservation(in both 3D and depth-integrated forms), momen-tum conservation, and conservation of salt andheat:
@u
@xþ
@v
@yþ
@w
@z¼ 0; (1)
@Z@t
þ@
@x
Z HRþZ
HR�h
udz þ@
@y
Z HRþZ
HR�h
vdz ¼ 0; (2)
Du
Dt¼ fv �
@
@xgðZ� acÞ þ
Pa
r0
� ��
g
r0
Z HRþZ
z
@r@x
d z
þ@
@zKmv
@u
@z
� �þ Fmx; ð3Þ
TED PROOF
Dv
Dt¼ � fu �
@
@ygðZ� acÞ þ
Pa
r0
� ��
g
r0
Z HRþZ
z
@r@y
d z
þ@
@zKmv
@v
@z
� �þ Fmy; ð4Þ
DS
Dt¼
@
@zKsv
@S
@z
� �þ F s; (5)
DT
Dt¼
@
@zKhv
@T
@z
� �þ
_Q
r0Cp
þ Fh; (6)
where
(x, y) horizontal Cartesian coordinates, (m)ðf; lÞ latitude and longitudez vertical coordinate, positive upward, (m)t time, (s)HR z-coordinate at reference level (geoid or
mean sea level (MSL))Zðx; y; tÞ free-surface elevation, (m)hðx; yÞ bathymetric depth, (m)~uð~x; tÞ water velocity at ~x=(x,y,z), with Carte-
sian components (u,v,w), (m s�1)f Coriolis factor, (s�1) (Section 2.5)g acceleration of gravity, in (m s�2)cðf; lÞ tidal potential, (m) (Section 2.5)a effective Earth elasticity factor (E0.69;
Foreman et al., 1993)rð~x; tÞ water density; by default, reference value
r0 is set as 1025 kgm�3
Paðx; y; tÞ atmospheric pressure at the free surface,(Nm�2)
S, T salinity and temperature of the water(practical salinity units (psu), 1C)
Kmv vertical eddy viscosity, (m2 s�1)Ksv, Khv vertical eddy diffusivity, for salt and heat,
(m2 s�1)Fmx;F my;F s;F h horizontal diffusion for momen-
tum and transport equations_Qðf; l; z; tÞ rate of absorption of solar radiation
(Wm�2)Cp specific heat of water (J kg�1K�1)
In the remainder of this text and in mostELCIRC simulations, we neglect horizontal diffu-sion in the momentum and transport equations.Horizontal diffusion tends to be a secondary
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process, and the solution method introducesnumerical horizontal diffusion that may exceedits physical counterpart (see further discussion inSections 3.4 and 4.1.3). The reader is referred toCasulli and Zanolli (1998) and Casulli and Cheng(1992) for a treatment of horizontal diffusionterms in the framework of Eulerian–Lagrangianfinite volume models.
The differential system for the six primaryvariables ðZ; u; v;w;T ;SÞ; Eqs. (1)–(6), is closedwith the equation of state (density as a function ofsalinity and temperature; Section 2.2), the defini-tion of the tidal potential and Coriolis factor(Section 2.5), parameterizations for vertical mixing(Section 2.4), and appropriate initial and bound-ary conditions. Initial conditions require problem-dependent specification of pre-simulation fields ofall primary variables and of any turbulenceparameters required by the vertical mixing para-meterization. We will discuss the vertical boundaryconditions in Section 2.3. Lateral boundary con-ditions may be chosen from a range of Dirichlet,Neumann and open boundary conditions.
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UNCORREC2.2. Equation of state
The density of sea water is defined as a functionof salinity, temperature and hydrostatic pressure,using the International Equation of State of SeaWater (ISE80) standard described in Millero andPoisson (1981):
rðS;T ; pÞ ¼rðS;T ; 0Þ
1� 105p=KðS;T ; pÞ� � ; (7)
where rðS;T ; 0Þ (kg/m�3) is the density at onestandard atmosphere, and KðS;T ; pÞ is the secantbulk modulus. Polynomial expressions for bothrðS;T ; 0Þ and KðS;T ; pÞ are integral to the ISE80standard, and will not be repeated here. Wecompute the water pressure in bars, consistentwith the hydrostatic approximation:
p ¼ 10�5g
Z HRþZ
z
rðS;T ; pÞdz: (8)
TED PROOF
2.3. Vertical boundary conditions for primary
equations
2.3.1. Horizontal momentum: surface boundary
At the sea surface, we enforce the balancebetween the internal Reynolds stress and theapplied shear stress, i.e.
r0Kmv
@u
@z;@v
@z
� �¼ ðtWx; tWyÞ at z ¼ HR þ Z:
(9)
ELCIRC allows for two different approaches tothe parameterization of spatially and temporallyvariable surface shear stresses. One approachconsists of the use of a bulk aerodynamicalgorithm developed by Zeng et al. (1998) toaccount for ocean surface fluxes (momentum, heatand salt) under various conditions of stability ofthe atmosphere. This approach is summarized inAppendix A, and is recommended when ELCIRCis used in conjunction with (or, more commonly,forced by outputs from) an atmospheric model.Short of detailed information on atmospheric
stability, surface stresses can alternatively beevaluated as
ðtWx; tWyÞ ¼ raCDsj ~W jðW x;W yÞ (10)
where, ra is the air density (kgm�3), CDs the winddrag coefficient (�), ~W ðx; y; tÞ the wind velocity at10m above the sea surface, with magnitude |W|and components Wx and Wy (ms�1)and where:
CDs ¼ 10�3ðAW1 þ AW2j ~W jÞ
if W lowpj ~W jpW high ð11Þ
with CDs held constant (at either Wlow or Whigh
values) outside the range, as appropriate. Formoderately strong winds, this formula allows theefficiency of the air–ocean transfer of momentumto increase with increasing wind speed. Manyalternative literature values have been proposedfor AW1, AW2 and associated ranges of validity(e.g., see review in Pond and Pickard (1998), pp.135–137). In the absence of data to the contrary,ELCIRC assumes, as the starting point for site-specific calibration, that
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UNCORREC
AW1 ¼ 0:61;
AW2 ¼ 0:063;
W low ¼ 6;
W high ¼ 50: ð12Þ
2.3.2. Horizontal momentum: bottom boundary
As customary, we enforce at the sea bottom thebalance between the internal Reynolds stress andthe bottom frictional stress, i.e.
r0Kmv
@u
@z;@v
@z
� �b
¼ ðtbx; tbyÞ; at z ¼ HR � h;
(13)
where the bottom stress is defined as
ðtbx; tbyÞ ¼ r0CDb
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2
b þ v2b
qðub; vbÞ: (14)
The bottom drag coefficient CDb is typicallyvariable in space, and might also vary at varioustemporal scales (e.g., through current–wave inter-actions or long-term changes in bottom texture).Site-specific calibration is often required. InELCIRC, the bottom drag coefficient can beexternally specified, or can be evaluated internallyby matching velocities (ub,vb) at or near the edge ofthe bottom boundary layer:
CDb ¼ max1
kln
db
z0
� ��2
;CDbmin
( ); (15)
where k ¼ 0:4 is the von Karman’s constant, z0 thelocal bottom roughness, and db half the thicknessof the bottom computational cell. The parameterz0 depends on the local bottom roughness, and istypically of the order of 1 cm (Blumberg andMellor, 1987). A coarse bottom discretization maygreatly overestimate db relative to the trueboundary layer thickness and, without the moder-ating effect of CDbmin, could grossly underestimateCDb. Values of CDbmin of 0.0075 and 0.0025 havebeen recommended for continental shelves (Lynchet al., 1996) and deep ocean (Blumberg andMellor, 1987), respectively, corresponding to an‘‘effective db’’ of approximately 1 and 30m.Ultimately, the choice is site-specific and spatiallyvariable.
OF
2.3.3. Heat and salt conservation
In most cases, there are no salt fluxes across thesea surface and bottom, neither is there any heatflux at the bottom. However, heat exchangesthrough the air–sea interface are important inmost coastal and ocean systems. While solarradiation is treated directly in Eq. (6), all otherheat exchanges must be accounted through thesurface boundary condition. Specifically:
Khv
@T
@z¼
H�tot #
r0Cp
; at z ¼ HR þ Z; (16)
where H�tot # is the net downwards heat flux at the
air–water interface, exclusive of solar radiation(see Appendix A).
TED PRO2.4. Parameterization of turbulent vertical mixing
Recognizing that the parameterization of turbu-lent vertical mixing remains an open question incoastal modeling, we allow for multiple choicesamong many approaches of widely varying com-plexity that have been proposed in the literature.Currently coded are a zero-equation model (basedon Pacanowski and Philander, 1981) and multipletwo-and-a-half equation models (Umlauf andBurchard (2003) and Mellor and Yamada (1982),as modified by Galperin et al. (1988)). In allapproaches, we assume similarity of the verticalmixing for heat and salt, i.e.Ksv ffi Khv:
2.4.1. Zero-equation models
Several zero-equation parameterizations at-tempt to account for the effect of varyingstratification on the vertical mixing. Vaz andSimpson (1994) compared three such schemes(Munk and Anderson, 1948; Pacanowski andPhilander, 1981; Lehfeldt and Bloss, 1988) amongeach other and against simpler (constant and step-function viscosities) and more complex approaches(one- and two-equation schemes, from Mellor andYamada, 1982). The comparison, set up in acontext of transient estuarine stratification pro-blems and based on qualitative and quantitativetests, found the scheme of Pacanowski andPhilander (1981) to perform the best amongzero-equation closures.
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ORREC
The scheme assumes that the local eddyviscosity and diffusivity, Kmv and Khv, only dependon the gradient Richardson number, Ri. Specifi-cally,
Kmv ¼n0
ð1þ 5RiÞ2þ nb; (17)
Khv ¼Kmv
1þ 5Riþ Kb; (18)
where Kmv and Khv approach a turbulent uppervalue,n0; in the limit of no density stratificationand finite vertical shear (i.e., Ri ! 0), andapproach molecular background values,nband Kb,in the limit of large density stratification (i.e.,Ri ! 1). While Pacanowski and Philander (1981)recommended n0 ¼ 5� 10�3; nb ¼ 10�4 and Kb ¼
10�5 m2 s�1; ELCIRC gives the user the choice onthese values. In the above equations, the Richard-son number is defined as
Ri ¼N2
@u=@z� 2
þ @v=@z� 2 ; (19)
where the Brunt–Vassala frequency, N2; can benegative:
N2 ¼g
r0
@r@z
: (20)
2.4.2. Two-and-a-half equation models
We have implemented in ELCIRC both thetraditional 2.5 closure model of Mellor andYamada (1982) as modified by Galperin et al.(1988) (hereafter, MY25), and the generic lengthscale (GLS) closure model proposed by Umlaufand Burchard (2003). GLS includes, as particularrealizations, a variety of two-and-a-half equation
UNCTable 1
Constants for various GLS closure models
c p m n sck sc
k2� �e 3 1.5 �1 1.0 1.3
k2k‘ k‘ 0 1 1 1.96 1.96
k2o �ffiffiffik
p=‘ �1 0.5 �1 2.0 2.0
UB � k=‘2=3 2 1 �0.67 0.8 1.07
aValues reflect the choice of stability functions from Kantha and C
TED PROOF
models, both new (e.g., GLS as optimized byUmlauf and Burchard (2003), henceforth UB) andtraditional—such as k2� (kinetic energy andenergy dissipation, Rodi (1984)) and k2o (kineticenergy and frequency of dissipation, Wilcox,1998). Although the GLS framework does notstrictly include MY25, it allows for an analog(k � k‘; kinetic energy and kinetic energy timeslength scale).Central to the GLS framework are two equa-
tions that govern the transport, production anddissipation of the turbulent kinetic energy (k) andof a generic length-scale variable (c):
Dk
Dt¼
@
@znck
@k
@z
� �þ KmvM2 þ KhvN2 � �; (21)
DcDt
¼@
@znc
@c@z
� �þ
ckðcc1KmvM2
þ cc3KhvN2 � cc2F w�Þ; ð22Þ
where cc1; cc2 and cc3are model-specific constants(Table 1), Fw is a wall proximity function, M andN are shear and buoyancy frequencies, and e is thedissipation rate. The following definitions apply
M2 ¼@u
@z
� �2
þ@v
@z
� �2
; � ¼ ðc0mÞ3k1:5þm=nc�1=n
(23)
with the constant c0m set atffiffiffiffiffiffiffi0:3
p:
The generic length scale is defined as
c ¼ ðc0mÞpkm‘n; (24)
where the choice of the constants p, m and n
determines the specific closure model (Table 1).The desired vertical viscosities and diffusivities arerelated to k, c; and stability functions, in the form:
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cc1 cc2 cþc3 c�c3a Fw
1.44 1.92 1U0 �0U629 1
0.9 0.5 1U0 0.9 Eq. (29)
0.555 0.833 1U0 �0U642 1
1 1.22 1U0 0.05 1
layson (1994).
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UNCORREC
Kmv ¼; cmk1=2‘; Khv ¼ c0
mk1=2‘;
nck ¼Kmv
sck; nc ¼
Kmv
sc; ð25Þ
where nck and nc: are vertical turbulent diffusivitiesfor k and c; and the Schmidt numbers sck and scare model-specific constants (Table 1). An alge-braic stress model (e.g. Kantha and Clayson, 1994;Canuto et al., 2001; or Galperin et al., 1988) isrequired to define the stability functions. UsingKantha and Clayson (1994), the stability functionsassume the form
cm ¼ffiffiffi2
psm c0m ¼
ffiffiffi2
psh; (26)
with
sh ¼0:4939
1� 30:19Gh
; sm ¼0:392þ 17:07shGh
1� 6:127Gh
;
Gh ¼Gh u � ðGh u � Gh cÞ
2
Gh u þ Gh0 � 2Gh c
ð27Þ
and
Gh u ¼ min Gh0;max �0:28;N2‘2
2k
� �� �;
Gh0 ¼ 0:0233; Gh c ¼ 0:02: ð28Þ
The wall proximity function, Fw, allows modelswith a positive n (such as the MY25 analog, k2k‘)to satisfy boundary conditions. Fw is trivially unityfor models with a negative n. For k2k‘;
Fw ¼ 1þ 1:33‘
kdb
�2þ 0:25
‘
kds
�2; (29)
where db and ds are, respectively, the distance tothe computational bottom and to the sea surface:
dbðx; yÞ ¼ z � ½HR � hðx; yÞ�; (30)
dsðx; yÞ ¼ HR þ Zðx; yÞ � z: (31)
Table 1 shows the choices of constants p;m andn associated with four common GLS options, theresulting form of c; and the values of the constantscc1; cc2; cc3; s
ck and sc: Note that different values
of the buoyancy parameter cc3 are used for stableðc�c3Þ and unstable (cþc3) stratification. It is im-portant to recognize that, for all models, limits onlength scales are applied at several stages of thecalculations, to ensure that mixing remains posi-
TED PROOF
tive. The choice of these limits is practicallyimportant, yet theoretically ambiguous. While wein general follow the recommendations of Warneret al. (2004), we deviate from them in adopting thesame lower bound for c across all models, ratherthan model-specific bounds. Specifically, we usecmin ¼ 10�8 (the value recommended by Warner etal. (2004) for k2k‘). We found this common lowerbound for c important to ensure consistencyacross closure models, in the context of thesolution of selected benchmarks (e.g., Section 4.3).Regardless of the specific closure model, the
solution of Eqs. (21) and (22) requires boundaryconditions at the free surface and the computa-tional bottom. Rather conventionally, theseboundary conditions specify turbulent kineticenergy as a function of the frictional velocities atthe appropriate surface, in the form:
k ¼16:62=3
2u2� ¼
16:62=3
2Kmv
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@u
@x
� �2
þ@v
@y
� �2s
;
(32)
and specify c (note Eq. (24)) by additionallysetting the mixing length to the distances to thefree surface
‘ ¼ kdb or kds: (33)
We stated earlier that MY25 is not considered aspecial case of GLS. Consistent with this view, theELCIRC implementation of MY25 is ratherconventional (Mellor and Yamada (1982), asmodified by Galperin et al. (1988)), and is codedindependently of GLS. Yet, it is important torecognize that the main difference between MY25and GLS is arguably in details of the internallyimposed length-scale limits: in MY25 the lengthscale is only limited in the calculation of stabilityfunctions, while in GLS the length scale is limitedin the calculation of production, wall proximityfunction, and stability functions (Warner et al.,2004). Other significant differences between MY25and GLS include the stability functions (Galperinet al. (1988) versus multiple options, such as ourchoice of Kantha and Clayson (1994)) and thebuoyancy parameter cc3 (constant at 0.9 versusdependent on stratification stability, Table 1).
T
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2.5. Definition of Coriolis factor and tidal potential
Rather conventionally, the earth rotation isrepresented through the Coriolis acceleration inthe momentum equations. The Coriolis factor, f, isa well known function of latitude,f:
f ðfÞ ¼ 2O sin f; (34)
where O ¼ 7:29� 10�5 rad s�1 is the angular velo-city of rotation of the earth. To minimizecoordinate inconsistencies (Cartesian in ELCIRC,spherical in Eq. (34)), we use a b-plane approxima-
tion for f:
f ¼ f C þ bCðy � yCÞ; (35)
where subscript C denotes the mid-latitude of thedomain and b is the local derivative of the Coriolisfactor.
The tidal potential is defined following Reid(1990):
cðf; l; tÞ ¼Xn;j
Cjnf jnðt0ÞLjðfÞ
cos2pðt � t0Þ
Tjn
þ jlþ njnðt0Þ
�; ð36Þ
where
Cjn constants (e.g., Reid (1990)) characteriz-ing the amplitude of tidal constituent n ofspecies j (j=0, declinational; j=1, diurnal;j=2, semi-diurnal), (m)
t0 reference timefjn(t0) nodal factorsnjn(t0) astronomical arguments, (r)LjðfÞ species-specific coefficients ðL0 ¼
sin2 f; L1 ¼ sinð2fÞ; L2 ¼ cos2 fÞTjn period of constituent n of species j
N 89
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U3. Numerical algorithm
3.1. Overview
The numerical algorithm of ELCIRC followsthe functional sequence illustrated in Fig. 2, andhas the following important features:
1.
A semi-implicit scheme is used: (a) the baro-tropic pressure gradient in the momentumequation and the flux term in the continuityequation are treated semi-implicitly, with im-plicitness factor 0:5pyp1; (b) the verticalviscosity term and the bottom boundary condi-tion for the momentum equations are treatedfully implicitly; and (c) all other terms aretreated explicitly. This ensures both stability(Casulli and Cattani, 1994) and computationalefficiency.2.
ROOFThe normal component of the horizontalmomentum equations is solved simultaneouslywith the depth-integrated continuity equation,i.e., there is no mode splitting between theseequations. The total derivatives of the normalvelocity are discretized using Lagrangian back-tracking, thus preventing advection from im-posing stability constraints on the time step.
3.
PThe vertical velocity is solved from the 3Dcontinuity equation using a finite volumeapproach.4.
EDThe tangential component of the horizontalmomentum equations is formally solved withfinite differences. The solution is computation-ally efficient, because we re-use matrices formedand inverted in the process of computingnormal velocities.
5.
Once the full 3D velocity is recovered, thetransport equations for salinity and tempera-ture are solved at both the polygonal vertices(nodes) and centers of element sides, using finitedifferences. This amounts to splitting eachelement in the flow grid into four transportsub-elements, and reduces numerical diffusion.The solution requires backtracking along char-acteristic lines, which is done anew (i.e., withoutre-using the backtracking for normal velocities)to account for the most recent flow field. Afterthe salinity and temperature are found, thedensity is calculated from the equation of state,and is fed back to the momentum equations atthe next time step (i.e., the baroclinic term istreated fully explicitly).6.
If a two-and-a-half equation turbulence closureis invoked, the eddy viscosity and diffusivity arecomputed at each time step prior to the solution
ECTED PROOF
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or hot start
< < 1%
Back trackingfor momentum
~12% (4.7s)
TurbulenceClosure
MY25:~14% (5.5s).
2D continuityand3Dmomentum: set-up
Back substitution forhorizontal velocity
<< 1% Update free surface< < 1%3D continuityfor
vertical velocity< < 1%
Back tracking forscalar transport
~ 33% (12.8s)
Scalar transport forsalt & temperature
~ 5% (2.1s)
~ 5% (2.88swhen called for) ?
?
End
y
n
y
n
i=i+1Time step i
(including fileoutput)Input of forcings
~13.7% (53s)
Output to file
2D continuity:solver~ 1% (0.42s)
Tota CPU:386s
Initialization
Fig. 2. Functional sequence of the ELCIRC solution, and percentages of CPU used by major tasks. Note: Relative CPU varies with
the specific application. Numbers shown are for a time step with file output, within a recent river-to-ocean simulation of 3D baroclinic
Columbia River circulation (Baptista et al.)), in a single processor of a dual-processor 2.4GHz, 4Gb, Intel Xeon. The grid has 33,634
horizontal nodes, 50,389 horizontal elements and 62 vertical layers, for a total of E2.2M active prism faces.
3The choice of z-coordinates enables a natural treatment of
wetting and drying, but creates a stair-case representation of the
bottom that limits the representation of the bottom boundary
layer. A generalized sigma coordinate (cf. Song and Haidvogel,
1994) is being considered as an option for ELCIRC, but has not
yet been implemented.
Y. Zhang et al. / Continental Shelf Research ] (]]]]) ]]]–]]]10
UNCORRof the momentum equation, using informationfrom the previous time step.
Features 1–3 follow Casulli and Zanolli (1998)closely, while features 4–6 are deviations fromTRIM/UnTRIM strategies that we found useful incontrolling numerical diffusion (feature 5, cf.Section 4.1.3), improving the representation ofCoriolis (feature 4, cf. Sections 4.2.1 and 4.2.2) andproviding qualitatively realistic representations ofplume dynamics (features 4–6, cf. Section 4.4).
3.2. Domain discretization
The 3D domain is discretized into a series oflayers in the vertical and into a combination of
triangular and/or quadrangular elements in thehorizontal (Fig. 3a). Unstretched z-coordinates areused,3 with each layer extending throughout theentire horizontal domain and being numberedsequentially upwards. The thickness of the kthlayer (i.e., the distance between levels k�1 and k) isDzk; and the distance between half-levels isDzkþ1=2 ¼ ðDzk þ Dzkþ1Þ=2: Note that the thick-nesses of the bottom and top layers include onlythe portions occupied by water.
CORRECTED PROOF
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(u,v)S,Tk,l
S,T
S,T
S,T
(u,v)S,Tk,l
(u,v)S,Tk,l
η,w
w
4
3
21
(a)
(b) (c)
Fig. 3. ELCIRC grids are unstructured in the horizontal, with combinations of triangles and quadrangles (a), and use unstretched z-
coordinates in the vertical. As shown in (b), most variables (horizontal velocities, salinity, temperature and turbulent quantities) are
defined at half levels, either (or both) at nodes or side centers; water levels and vertical velocities are however defined at full levels, and
element centers. The definition of salinity and temperature at both nodes and side centers effectively means that horizontal elements are
split into four sub-elements for the purpose of solving the scalar-transport equations; for instance, interpolation at the foot of the
characteristic line shown in (c) is based only on sub-element 1.
Y. Zhang et al. / Continental Shelf Research ] (]]]]) ]]]–]]] 11
UNThe combined horizontal and vertical discreti-zations result in the whole 3D domain beingdivided into a series of prisms. The depths at eachside, calculated from depths at nodes, are assumedto be constant, and the depths in each element aretaken to be the maximum of depths at its sides.This results in a staircase representation of thebottom. Since the layer thicknesses at sides,elements, and nodes are in general different from
each other, superscripts ‘‘e’’ and ‘‘p’’ are added toDz to denote the elements and nodes, while Dz isreserved for sides. Some of the other notations tobe used are:
Np number of nodes (vertices) in the hor-izontal grid
Nv number of levels in the vertical gridNe number of elements in the horizontal grid
O
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Ns number of sides in the horizontal gridjs(i,j) (j=1, y, i34(i)) sides of elementi34(i) number of sides in element i
is(j,i) (i=1,2) two elements that share side j
ip(j,i) (i=1,2) two end nodes of side j
lj length of side j
Pi area of element i
dj distance between the two element centerssharing side j
mj (mej ; m
pj ) bottom level index for side (element/
node) j
Mj (Mej ; M
pj ) free-surface level index for side
(element/node) j.
We generally use subscripts for spatial location,and double subscripts for horizontal and verticallevel indices.
A staggering scheme is used for the definition ofvariables (Fig. 3b). The elevation, defined atelemental centers, is assumed to be constant withineach element. The normal and tangential compo-nents of the horizontal velocity, which are theactual unknowns to be solved from the momentumequations, are defined at the center of each verticalface of prism (i.e., at element side centers on halflevels). The vertical velocity w is located at elementcenters on whole levels. The salinity, temperatureand density are defined at vertices and side centers,for the reason outlined below.
As discussed by Casulli and Zanolli (1998),orthogonality4 is, in a strict sense, a requirementfor calculation of finite difference approximationsof spatial gradients in unstructured grids. Thisrequirement might in practice be relaxed, but theaccuracy of solutions suffers from deviations fromorthogonality. While a second-order accuracy canbe achieved with uniform structured or unstruc-tured orthogonal grids, only first-order accuracy isattainable with non-uniform orthogonal grids. Forgeneral non-orthogonal grids, the line connectingthe two element centroids is not perpendicular to
U4Following Casulli and Zanolli (1998), a grid is defined as
orthogonal if within each element a point (‘‘center’’, although
not necessarily the geometric center) can be identified such that
the segment joining the centers of two adjacent elements, and
the side shared by the two elements, have a non-empty
intersection and are perpendicular to each other.
OF
the common side, which is an additional source oferrors.Grid generation packages, including those
typically used by the authors (Turner and Baptista(1991) and Zhang and Baptista (2000)), do notspecifically ensure grid orthogonality. The burdenof creating orthogonal or near-orthogonal grids isthus on the user, and is non- trivial for complexdomains. In unconstrained parts of a computa-tional domain, it is typically easy to generateorthogonal grids based on quadrangles. In moreconstrained regions, hybrid-element grids may beused to avoid major deviations from orthogonality(e.g., by selectively merging two non-orthogonaltriangles into a quadrangle, within a region wherethe grid is predominantly formed by triangles).
91
93
95
TED PR3.3. Solving the depth-integrated continuity and
horizontal momentum equations
A key to UnTRIM is that the solution of thedepth-integrated continuity equation and horizon-tal momentum equations is conducted using local(element-side based) coordinate systems. Follow-ing Casulli and Zanolli (1998), we locally re-orient(x,y) such that the x-axis points outside of elementis (j, 1), from the center of side j. Eqs. (3) and (4)are invariant under a rotation in the (x, y) plane,and thus they retain their form under these localrotations. However, these equations now representlocal conservation of normal and tangentialmomentum, respectively. For consistency, u andv refer from here onward to normal and tangentialvelocities.Also, following Casulli and Zanolli (1998), we
impose local (and thus global) volume conserva-tion by using a semi-implicit finite-volume ap-proach to integrate the continuity equation, Eq.(2), for element i:
PiðZnþ1i � Zn
i Þ þ yDtXi34ðiÞl¼1
si;l‘jsj
XMjsj
k¼mjsj
Dznjsj;kunþ1
jsj;k
þ ð1� yÞDtXi34ðiÞl¼1
si;l ljsj
XMjsj
k¼mjsj
Dznjsj;kun
jsj;k ¼ 0;
i ¼ 1; . . . ;Ne; ð37Þ
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where y is the implicitness factor for temporaldiscretization, jsj=js(i,l) and si,l is a sign:
si;l ¼isðjsj; 1Þ þ isðjsj; 2Þ � 2i
isðjsj; 2Þ � isðjsj; 1Þ: (38)
For stability reasons, a semi-implicit finite-difference scheme is used to solve the normalmomentum equation for a side j:
Dznj;kðu
nþ1j;k � u�
j;kÞ ¼ Dznj;kf jv
nj;k Dt � Dzn
j;k
gDt
dj
y Znþ1isðj;2Þ � Znþ1
isðj;1Þ
� hþð1� yÞ Zn
isðj;2Þ � Znisðj;1Þ
� i
� Dznj;k
gDt
r0dj
XMj
l¼k
Dznl rn
isðj;2Þ;l � rnisðj;1Þ;l
� "
�Dzn
j;k
2rn
isðj;2Þ;k � rnisðj;1Þ;k
� �
þ Dt ðKmvÞj;k
unþ1j;kþ1 � unþ1
j;k
Dznj;kþ1=2
"
�ðKmvÞj;k�1
unþ1j;k � unþ1
j;k�1
Dznj;k�1=2
#
þ Dznj;kDt
@
@xgac�
Pa
r0
�� �n
j;k
;
j ¼ 1; . . . ;Ns; k ¼ mj ; . . . ;Mj; ð39Þ
where u is the normal velocity, and u�j;k is the
backtracked value at time step n at the foot of thecharacteristic line (see Section 3.4). All the right-hand side terms except for the elevation gradientand the vertical viscosity terms are treated fullyexplicitly. The discretized form of the tangentialmomentum equation is similar.
The discretized continuity and momentumequations can be written in compact matrix formas
Anj Unþ1
j ¼ Gnj � y g
Dt
dj
Znþ1isðj;2Þ � Znþ1
isðj;1Þ
h iDZn
j (40)
Anj V
nþ1j ¼ Fn
j � ygDt
lj
Znþ1ipðj;2Þ � Znþ1
ipðj;1Þ
h iDZn
j (41)
Znþ1i ¼ Zn
i �yDt
Pi
Xi34ðiÞl¼1
si;l‘jsj DZnjsj
h iTUnþ1
jsj
�ð1� yÞDt
Pi
Xi34ðiÞl¼1
si;l‘jsj DZnjsj
h iTUn
jsj ; ð42Þ
where Gnj and Fn
j are vectors that combine all
TED PROOF
explicit terms (including the baroclinic forcing),and where
Unþ1j ¼
unþ1j;Mj
..
.
unþ1j;mj
26664
37775; Vnþ1
j ¼
vnþ1j;Mj
..
.
vnþ1j;mj
26664
37775; DZn
j ¼
Dznj;Mj
..
.
Dznj;mj
2664
3775:
(43)
Matrix A, after adjustment to include thevertical boundary conditions (Eqs. (9) and (13))and applicable horizontal boundary conditions,remains tri-diagonal, and thus can be invertedquite efficiently. Normal velocities can therefore beexpressed as
Unþ1j ¼ ½An
j ��1Gn
j � ygDt
dj
Znþ1isðj;2Þ � Znþ1
isðj;1Þ
h i½An
j ��1DZn
j
(44)
and substitution of Eq. (44) into (42) leads to a setof equations for elevations at all elements(1pipNe):
Znþ1i �
gy2 Dt2
Pi
Xi34ðiÞl¼1
si;l ljsj
djsj
DZnjsj
h iT
Anjsj
h i�1
DZnjsj Znþ1
isðjsj;2Þ � Znþ1isðjsj;1Þ
h i
¼ Zni �
ð1� yÞDt
Pi
Xi34ðiÞl¼1
si;l ljsj DZnjsj
h iT
Unjsj
�yDt
Pi
X3l¼1
si;l ljsj DZnjsj
h iTAn
jsj
h i�1
Gnjsj ; ð45Þ
from which elevations can be solved. As indicatedin Casulli and Zanolli (1998), the coefficient matrixresulting from the above system of equations issymmetric and positive definite, and thus efficientsparse matrix solvers like Jacobian ConjugateGradient can be utilized.At open boundaries, elevations may be specified,
elevations may be nudged to specified values, ortransmissive boundary types (e.g., Flather (1987))may be applied. Because strict transmissiveboundary conditions can be very involved forunstructured grids, a simplified form is used, wherethe open-boundary elevations are computed as theaverage over all adjacent non-open boundaryelements, assuming that the phase speed there is
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5Positivity requires that solutions remain bounded by the
initial maxima and minima, in the absence of external sources
and sinks (and, thus, implies the absence of numerical
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UNCORREC
equal to the average grid size divided by the timestep.
Once the elevations are known, normal velo-cities can be computed at face centers by using Eq.(44). UnTRIM converts these normal velocitiesdirectly to the horizontal velocities at each node,using purely geometric arguments. However, wefound this approach unsatisfactory in ELCIRC forbenchmarks where Coriolis is a significant factor(e.g., Sections 4.2 and 4.4). Instead, we solve forthe tangential momentum equation, Eq. (41), priorto mapping velocities at the nodes. Although moretime consuming than UnTRIM’s, this approach isstill computationally efficient. Indeed, it fully re-uses matrix A, as formed and inverted for thesolution of the normal momentum equation.
Integral to the approach is, however, knowingelevations at nodes, Z; so that pressure gradientsalong sides can be computed. We estimate nodalelevations from elevations at element centers,computed earlier through Eq. (45), by using theformula
Znþ1i ¼
RZdSRdS
¼
Pj
Pineði;jÞZnþ1ineði;jÞP
j
Pineði;jÞ; i ¼ 1; . . . ;Np;
(46)
where the integrations and summations are carriedout over a ‘‘ball’’ around node i, with contribu-tions from all surrounding elements, ineði; jÞ: Theformula can be argued to be algebraically con-sistent with volume conservation within the ball(not shown).
3.4. Backtracking and interpolation along
characteristic lines
Both in the momentum equations (Section 3.3)and in the equations of salt and heat conservation(Section 3.6), we avoid usual Courant numberconstraints by incorporating advection in totalderivatives, and solving the resulting equations inan Eulerian–Lagrangian context. Integral to thisapproach is the ability to backtrack characteristiclines efficiently and accurately (e.g., see Oliveiraand Baptista, 1998), starting from known locationsat time n+1. Once the location of the foot of the
TED PROOF
characteristic lines at time n is found, initialconditions at that time step can be obtained forthe variable of interest by either interpolation orintegration (Oliveira and Baptista, 1995).In practice, backtracking is the single most time-
consuming part of our solution, a problem that isaggravated by the fact that we backtrack twice:first for the momentum equation, and then—usingupdated flow fields and a larger set of character-istic lines—again for the scalar-transport equa-tions. Backtracking for momentum starts alwaysat side centers, while backtracking for salinity andtemperature starts both at side centers and nodes(see Section 3.6). The approach is the same,regardless of location. In all cases, backtrackingrequires the 3D solution, backwards from n þ 1 ton; of
dxi
dt¼ um
i ðx1;x2;x3; tÞ with i ¼ 1; 2; 3; (47)
where m either stands for time step n (formomentum) or denotes a linear interpolantbetween n þ 1 and n (for salinity and temperature).Hence, flow fields are always known beforehand.As a compromise between accuracy and com-
putational efficiency, we backtrack using a simpleEuler integration of Eq. (47), but with a time stepsmaller than Dt: The approach is illustrated in Fig.4, and represents a deviation from more elabo-rated strategies that we have used in 2D modelssuch as Baptista et al. (1984), Oliveira and Baptista(1995), Wood et al. (1995) and Oliveira et al.(2000). The reader is referred to Oliveira andBaptista (1998) for an analysis on how trackingerrors may destroy the positivity and massconservation of solutions of the transport equa-tion.Also, in a deviation from our prior models, we
chose linear interpolation at the feet of thecharacteristic lines for both momentum andscalar-transport equations. The defining advan-tage of linear interpolation is the positivity5 of thesolutions, a property that we found particularlyimportant in addressing the representation of
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t
Characteristic linet+∆
Fig. 4. Backtracking of the characteristic lines (illustrated here in a 2D setting) is a time-consuming operation. It involves, for each
‘‘origin’’ at time t þ Dt; the 3D solution of Eq. (47). A simple Euler method is employed, but multiple tracking sub-steps are allowed
within the overall time step Dt: The number of sub-steps is imposed by the user, but might be overridden by code-controlled
adjustments accounting for local flow gradients.
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baroclinic forcings (e.g., see discussion in Section3.6). The disadvantage is that linear interpolationintroduces numerical diffusion (Baptista (1987);also, Section 4.1.3). To reduce numerical diffusionin the solution of the salinity and temperature, wesub-split (Fig. 3c) the original grid elements intofour before interpolating (which is the reason whywe solve the salt and heat balance equations atboth nodes and side centers). We do not usesplitting to solve the momentum equations, whereadvection terms—albeit important—often do nothave as dominant a role; should the need arise, thehorizontal grid must thus be refined to reducenumerical diffusion.
3.5. Vertical velocity solution
Defined as a constant within each element, thevertical velocity can be calculated from the 3Dcontinuity equation through a finite-volume ap-proach:
wnþ1i;k ¼ wnþ1
i;k�1 �1
Pi
Xi34ðiÞj¼1
si;j ljsjDznjsj;kunþ1
jsj;k;
k ¼ mei ; . . . ;M
ei ð48Þ
Although only the bottom boundary conditionwnþ1
i;mei�1 ¼ 0 is needed in this recursive formula, due
to volume conservation the free-surface conditionis also automatically satisfied, within a very smallclosure error. This is expected because Eq. (2) is
TED PRO
derived by integrating Eq. (1) over depth withkinematic boundary conditions at the bottom andfree surface (cf., Luettich et al., 2002).While much smaller in magnitude than the
horizontal velocities, the vertical velocity stronglyaffects the stability of stratification in estuaries andplumes. In our experience in the Columbia Riverestuary, overestimation or parasitic oscillations inthe computation of vertical velocities can easilytransform a two-layer flow into a well-mixed flow.Because the vertical velocity is a measure of thehorizontal divergence, this effectively means thatthere is very little tolerance for numerical oscilla-tions on the horizontal velocities. A practicalconsequence is that, for strongly stratified flows,time steps must be selected consistently with thecelerity of internal baroclinic waves (see alsoSection 3.6).
3.6. Solving for salt and heat balances
The numerical solution for salinities and tem-perature is obtained in an Eulerian–Lagrangianframework by the finite-difference solution of Eq.(5) or (6), at both nodes or side centers. Anexample is given below, for the solution of salinityat a side center:
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Dzni;k Snþ1
i;k � S�i;k
� ¼ Dt ðKhvÞi;k
Snþ1i;kþ1 � Snþ1
i;k
Dzni;kþ1=2
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�ðKhvÞi;k�1
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#
ði ¼ 1; . . . ;Ns; k ¼ mi; . . . ;MiÞ;
ð49Þ
where S is defined at half levels and S�i;k is the value
at the foot of the characteristic lines. Very similarequations result for temperature, and for solutionsat nodes rather than side centers.
After backtracking and local interpolation,temperature and salinity are solved independentlyat each vertical (node or side center) via Eq. (49) orsimilar. In addition, Neuman-type boundary con-ditions are imposed at both the bottom and freesurface, consistently with Section 2.3.3. Horizon-tally, the salinity and temperature at openboundaries are most often specified during inflowand radiated freely during outflow (which thebacktracking allows for rather naturally).
We note that the continuity and momentumequations are coupled to the salt and heat balanceequations via the baroclinic pressure term. Thisterm, if treated improperly, can in stronglystratified regions (e.g. the Columbia River estuary)create noise in the horizontal flow field, leading tonoise in the vertical velocity field, with thepotential to quickly annihilate stratification. Keysto successfully handling baroclinic forcing appearto be (a) the use of time steps limited by the speedof propagation of internal baroclinic waves, ratherthan of barotropic waves; and (b) positivity-preserving strategies for transported quantities(Section 3.4).
It is also important to avoid underestimatingbaroclinic effects by overly diffusing the densityfield. Numerical diffusion in salt and temperaturetransport can be controlled by: using time stepsthat lead to Courant numbers larger than unity,thus reducing the number of required interpola-tions (cf., Eq. (55), Section 4.1.3); and enablingsufficient spatial resolution (including sub-split-ting, Section 3.4) to resolve physically importantgradients.
TED PROOF
3.7. Parameterizing turbulence
The momentum and transport equations con-tain vertical viscosity and diffusivity that must beparameterized (Section 2.4). While no additionalequations must be solved with zero-equationclosures, both GLS and MY25 require twoadditional transport equations. We consider herethe case of GLS. The turbulent kinetic energy andmixing length are both defined at side centers andat half levels, and the approach to solving the twoclosure equations is similar to that for the scalar-transport equations (e.g., Eq. (49)). However,advection is neglected, and therefore no back-tracking is involved: the resulting equation isdirectly 1D in the vertical direction.Following Warner et al. (2004), we enhance
numerical stability through two procedures. First,the production terms are treated either explicitly orimplicitly depending on the sign. For example, theequation for the generic length-scale variable c;Eq. (22), is discretized as
Dznj;kðc
nþ1j;k � cn
j;kÞ
¼ Dt ðncÞnj;k
cnþ1j;kþ1 � cnþ1
j;k
Dznj;kþ1=2
"�ðncÞ
nj;k�1
cnþ1j;k � cnþ1
j;k�1
Dznj;k�1=2
#
þ M � DtDznj;kcc2½Fwðc
0mÞ
3k1=2‘�1�nj;kcnþ1j;k ;
j ¼ 1; . . . ;Ns; k ¼ mj ; . . . ;Mj ; ð50Þ
where the production term is treated as:
M ¼
DtDznj;kðcc1KmvM2 þ cc3KhvN2Þ
nj;k
cnþ1j;k
knj;k; if Mp0;
DtDznj;kðcc1KmvM2 þ cc3KhvN2Þ
nj;k
cnj;k
knj;k; if M40:
8><>:
ð51Þ
Secondly, the boundary conditions for the twoGLS equations are recast into flux form:
nck@k
@z¼ 0; at z ¼ HR � h or z ¼ HR þ Z;
(52)
nc@c@z
¼ knncc‘; at z ¼ HR � h; (53)
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nc@c@z
¼ �knncc‘; at z ¼ HR þ Z: (54)
Only after k and c are found with the aboveboundary conditions, are the original boundaryconditions, Eqs. (32) and (33), enforced.
3.8. Solving for wetting and drying
One of the major advantages of the formula-tions of Casulli and Cheng (1992) and of Casulliand Zanolli (1998) is their natural and robusthandling of wetting and drying. We retain theirapproach in ELCIRC, in what amounts toprimarily careful bookkeeping of indices. Afterall unknowns have been found for time step n+1,the free-surface indices are updated with the newlycomputed elevations. Elements are dried if h þ
Zoh0 (a small positive number, h0; is used in thecode in lieu of zero in order to avoid underflow). Itis also noteworthy that in the limit of only onevertical layer, the above formulation and numericsautomatically reduce to the 2D depth-integratedversion.
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Controlled numerical benchmarks have beenvery useful in understanding and enhancingELCIRC. A sub-set of these benchmarks isdescribed below, organized under four themes:basic numerical properties, Coriolis representa-tion, representation of stratification, and repre-sentation of qualitative plume behavior.
4.1. Basic numerical properties
Similarly to TRIM and UnTRIM, the under-lying numerical algorithm of ELCIRC is welladapted to large Courant numbers, and is volumeconserving, positivity preserving, low order, andnumerically diffusive (cf., Eq. (55)). The reader isreferred to Casulli and Cattani (1994) for con-textual formal analysis of the stability andaccuracy of this type of method. Also useful isthe extensive body of literature available onEulerian–Lagrangian methods for scalar trans-
TED PROOF
port, including Baptista (1987), Oliveira andBaptista (1995), Oliveira and Baptista (1998),and Oliveira et al. (2000). In the following sub-sections, we use simple benchmarks to illustratethe key numerical properties of ELCIRC. Furtherdetails on these and other benchmarks, includinginput files to reproduce benchmark results, areavailable electronically (CCALMR, 2003).
4.1.1. Volume conservation
Volume conservation is algorithmically enforcedthrough the use of a finite-volume strategy. Inpractice, the testing of early versions of ELCIRCrevealed that substantial deviations from strictvolume conservation might result from ambiguityin coding choices on the treatment of inter-elementdiscontinuities and wetting and drying. However,attention to coding detail can virtually eliminatevolume conservation errors. New versions ofELCIRC are now tested for volume conservationthrough a benchmark that involves moving a fixeddischarge along an irregular riverbed, with bothhorizontal and vertical complexity (Fig. 5a). Thisbenchmark is inspired on the need of cross-scalemodels to correctly propagate freshwater volumesand rates through river networks to estuaries andultimately the ocean. The desired result of thebenchmark is to exactly match inflow and outflowdischarges, at equilibrium. As illustrated in Fig.5b, ELCIRC_v5.01 inflow and outflow dischargesat control transects do indeed match. Numerically,the match is within 0.002% of the equilibriumdischarge.
4.1.2. Implictness factor
Casulli and Cattani (1994) and Casulli andZanolli (1998) showed that the judicious choice ofimplicit or explicit treatment for each term of themomentum equation is critical for the computa-tional performance, algorithmic robustness andaccuracy of TRIM and UnTRIM (cf., Section 3.1).Casulli and Cattani (1994) further suggest asignificant effect of the choice of the implicitnessfactor, y; used for terms being treated semi-implicitly. In particular, stability requires0:5pyp1; and theoretical optimal accuracy isobtained for y ¼ 0:5; with numerical dampingincreasing progressively for larger y: While these
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Fig. 5. Volume conservation of successive ELCIRC implementations is systematically tested against the irregular riverine domain
shown in (a). Discharges are measured at control cross sections (transects 1 and 2). For several versions now ELCIRC results have
been consistently very good. Results for ELCIRC_v5.01 are shown in (b).
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UNCORREguidelines are strictly valid only under idealizedconditions, they tend to be broadly useful.
Here, we use a conventional benchmark toexamine the role of y on ELCIRC accuracy fordepth-averaged long-wave propagation over alinearly sloping bottom. The computational do-main is a quarter annulus with inner and outerradii of 60,960 and 152,400m (Fig. 6a). Waterdepths at the inner (land) and outer (ocean)boundaries are 10.02 and 25.05m, respectively.An M2 tide of amplitude 0.3048m propagatesfrom the open ocean to the slope and is refractedand reflected by the bottom and land boundaries.Water is considered inviscid, and the bottomfrictionless. Assuming linearity (a reasonableapproximation, given the choice of water depths
and tidal amplitudes), an exact solution exists forthe problem (Lynch and Gray, 1978).A convergence study, using several horizontal
grids, time steps and implicitness factors showedthat large time steps (and Courant numbers wellabove unity) may be used in ELCIRC for thisbarotropic problem without incurring instability,provided 0:5pyp1: Results are presented in Fig.6b for Dt ¼ 1047:93 s; and show the behavior oferror relative to the implicitness factor. Errormetrics shown are percentage errors relative to theanalytical solution of the amplitude of M2
elevations and radial velocities, at a node located240m from the inner boundary. Only the stabilityregion is shown. As anticipated by the analysis ofCasulli and Cattani (1994), accuracy degrades
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Fig. 6. Long-wave propagation over a linearly sloping bottom. The horizontal discretization of the quarter annulus domain is shown
in (a). The problem is solved for a single vertical layer. Results, shown in (b) in the form of percentage errors for the amplitudes of M2
elevation and radial velocity, illustrate that accuracy degrades progressively with the increase of the implicitness factor within the
stability region (0:5pyp1).
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UNCORRECprogressively from y ¼ 0:5 to 1, reflecting increas-ing numerical damping.
In practice, we often adopt y ¼ 0:6 to remainclose to optimal accuracy while eliminating orminimizing any oscillations introduced by thebaroclinic terms not included in the analysis ofCasulli and Cattani (1994).
4.1.3. Transport in an uniform flow
Eulerian–Lagrangian solutions of the transportequation have been extensively studied in theliterature and are commonly used in scientific andengineering applications. The single largest advan-tage of this class of methods is their ability tohandle Courant numbers larger than unity. Yet,solutions are not strictly mass preserving (Oliveiraet al., 2000), require accurate tracking of char-acteristic lines as a pre-condition for overallaccuracy and mass conservation (Oliveira andBaptista, 1998), and may exhibit substantialnumerical diffusion depending on the choice ofthe interpolation or integration approach used at
Tthe foot of the characteristic lines (Baptista, 1987;Oliveira and Baptista, 1995). As discussed earlier,we chose to implement in ELCIRC an adjustablyaccurate tracking algorithm for characteristiclines, to enable control over positivity preservationand mass conservation. Also, we use simple linearinterpolation at the foot of the characteristic lines,a choice driven by efficiency and positivityconsiderations but known to induce numericaldiffusion—thus placing on the user the onus ofchoosing an appropriate time step and spatialresolution. To partially mitigate numerical diffu-sion, ELCIRC splits—during the solution of thescalar-transport equations—each element of thenumerical grid into four sub-elements (Fig. 3c),which increases the dimensionless wavelength ofthe transported fields.In this section, we use the conventional problem
of advective transport of a scalar field in a uniformflow, to show that ELCIRC follows expectedbehavior relative to the theory of Eulerian–La-grangian methods. The reference simulation con-
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sists of advecting a Gauss hill, with standarddeviation 1 km and amplitude 5 1C (above abackground temperature of 5 1C), in a flow fieldconstant in space and time with u=0.1m�1 s andv ¼ 0: The problem was set up computationallywith a 30 km� 6 km rectangular channel of con-stant depth 20m, which was then uniformlymeshed with quadrangles with Dx ¼ Dy ¼ 250m:The correct solution is the transport of the Gausshill by the ambient flow without any deformation.The problem is essentially 1D in the x-direction.We can formally show that in this case theELCIRC algorithm has a diffusion-like leading-order truncation error, of the form
� ¼f 00
�
8Dx2 fracðCuÞ 1� fracðCuÞð Þ; (55)
where Cu ¼ 2uD t=Dx is the Courant number(defined to account for the splitting in sub-elements), fracðCuÞ is the fractional part of theCourant number, and f 00
� is the second derivativeof the solution at the foot of the characteristic line.
Fig. 7 synthesizes results of several ELCIRCsimulations. Fig. 7a shows that errors per time stepmimic the behavior predicted by Eq. (55). Fig. 7billustrates the diffusive nature of the underlyingalgorithm, and shows that the use of larger timesteps improves accuracy after a set period of time.This latter trend is common in Eulerian–Lagran-gian methods, the explanation being that errorsper time step depend on fracðCuÞ rather than Cu;and thus larger time steps imply less instances of asimilar error over a fixed period of time. Note that,in spite of this trend, Eulerian–Lagrangian meth-ods are numerically consistent, with the truncationerrors approaching zero when both Dx and Dt tendto zero (Baptista, 1987). Finally, Fig. 7c is areminder that the method is only first order inspace.
4.2. Coriolis representation
Coriolis plays a major role in coastal dynamics,and its correct representation is a critical require-ment for cross-scale, river-to-ocean circulationmodels. While early versions of ELCIRC wereunable to meet this requirement, the problem wasovercome by the inclusion of the momentum
TED PROOF
equation for the transversal velocities (Section3.3), one of the significant deviations of ELCIRCrelative to UnTRIM. We present below results ofELCIRC_v5.01 for two benchmarks that wereinstrumental in identifying and remedying theinitial problems.
4.2.1. Geostrophic flow in a straight channel
In the presence of Coriolis, flow in a constrainedchannel develops a lateral slope normal to themean flow direction. A steady-state analyticalsolution for an inviscid fluid can be found bybalancing the pressure gradient with Coriolis(Pond and Pickard, 1998). Matching this analy-tical solution requires correct representation ofCoriolis. We consider here a 20 km� 1 km rectan-gular channel of constant depth 20m, with theright end linked to a large deep basin(20 km� 5 km� 200m) to reduce downstreamboundary effects. We place the channel at 451Nlatitude (i.e., f � 10�4 s�1), and impose a constantflow of 1.98� 104m3 s�1 on the left end of thechannel. Surface elevations on the basin end arekept at MSL. Both a uniform triangular grid(Dx=125m; Dy=2km) and moderate non-uni-form variations thereof were tested, with essen-tially identical results. The vertical resolution Dz
varies from 5 to 100m, and the time step is set at5min. A steady state is established shortly after a1-day ramp-up. Away from the boundary, the flowis essentially uniform inside the channel (Fig. 8a,b), with a maximum error in the computed velocityof about 1 cm/s (or 1% of the theoretical velocityof 1m s�1). A nearly linear elevation slope isestablished normal to the flow, and compares wellwith the analytical solution (Fig. 8c).
4.2.2. Ekman dynamics
Away from the equator, wind blowing over theocean leads to circulation patterns of majoroceanographic and ecological relevance. Ekman
dynamics (e.g., see Pond and Pickard, 1998)describes the expected behavior for a steady windblowing over an infinitely deep and wide oceanwith constant density, assuming a balance betweenfriction (wind stress and vertical eddy viscosity)and Coriolis. The analytical solution showscurrents at an angle with the direction of the
ORRECTED PROOF
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Fig. 7. Advection of a Gauss Hill in an uniform flow. (a) Errors per time step depend on the fractional part of the Courant number, as
anticipated by Eq. (55). Results shown are for a fixed Dx ¼ 250m; and variable Dt: (b) Numerical damping at a fixed time increases
with decreasing time step. Results shown are for a fixed Dx ¼ 250m: (c) Errors as a function of Dx; showing first-order convergence.
Results shown are for a fixed Dt ¼ 500 s:
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UNCwind: 45o
to the right of the wind direction at thesurface, in the northern hemisphere, and rotatingwith depth in a spiral pattern known as the Ekman
spiral. At the Ekman depth, the direction of thewind-driven current is exactly opposite to itsdirection at the surface. The velocity of Ekman
currents decays exponentially with depth, fromabout 1.5% of the wind speed at the sea surface to0.06% at the Ekman depth.
We describe in this section ELCIRC_5.01solutions for an Ekman dynamics benchmarkproblem. We consider a square domain of100 km� 100 km with a flat (50m in depth) andfrictionless bottom (to approximate an infinitedepth) at about 451N latitude (fE10�4 s�1). Theuniform wind is set at 10m s�1, and the constantvertical eddy viscosity at 10�4m2 s�1. The hor-izontal resolution is uniform with Dx ¼ Dy ¼
2 km; and Dz varies from 0.5 at the surface to
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Fig. 8. Geostrophic flow in a straight channel, for a uniform grid. (a) Isolines of elevation after 5 days. (b) Cross-sectional velocity
magnitudes, at x=10km (exact solution is 1m s�1). (c) Comparison of analytical and numerical solutions for elevation at x=10km.
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UNCO27m at the bottom (we have also conducted testswith moderately non-uniform grid variations, withessentially the same results). The elevation isclamped at the MSL on the four edges of thesquare domain. The time step is set at 5min. Thenon-linear advection terms are turned off tofacilitate comparison with the analytical solution.To study the sensitivity of the solution with respectto the wind direction, we applied two winddirections of 901 (‘‘North’’) and 601 to thehorizontal.
A quasi-steady state, with small inertial pertur-bations, is established shortly after the ramp-upperiod (which is 1 day in this case), with uniformvelocity at each layer (except for slight variationsnear the boundary). Other than a rigid-bodyrotation, the solution is not sensitive to the choiceof wind direction, thus suggesting the correcttreatment of Coriolis in the numerical scheme. Thevertical profiles of velocity at the center of thedomain, averaged over 3 days after ramp-up, arecompared with the analytical solution (Fig. 9a–d).The comparison is generally excellent inside the
RRECTED PROOF
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Fig. 9. Ekman dynamics. Comparison of analytical and steady-state ELCIRC solutions at the center of a square domain, under
alternative wind directions. (a) Velocity magnitude. (b) Error in velocity magnitude. (c) Velocity direction. (d) Error in velocity
direction.
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UNCOEkman layer (below the Ekman depth the velocityis very small and the comparison between thenumerical and analytical solutions has no practicalsignificance). Errors increase right at the surface ofthe Ekman layer, an artifact of the representationof normal and tangential velocities at half levels(rather than exactly at the surface).
4.3. Adjustment under gravity
Stratification plays an important role in thedynamics of estuarine and marine systems. A
demanding benchmark for the ability of circula-tion models to represent stratification involves thegravitational adjustment of two fluids of differentdensity, initially separated by a vertical wall(Wang, 1984). Once the vertical wall is removed,the fluids adjust through two density frontstraveling in opposite directions, to eventually forma stably stratified two-layer flow. While there is nostrict analytical solution for this problem, modelssuch as ROMS and SEOM have been extensivelytested against it, with simulations evaluated bothrelative to qualitative behavior (e.g., front sharp-
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ness and solution smoothness) and quantitativemetrics (celerity of propagation of density fronts;and maximum and minimum density).
The ELCIRC_v5.01 solution of the inviscidform of this problem is shown in Fig. 10a, withdiscretization and parameter choices described inthe caption. Associated quantitative metrics showthat the expected maxima and minima of densityare exactly preserved—a consequence of positiv-ity—but that the speed of the gravitationaladjustment is about 84% of that predicted bylinear theory. Increasing the vertical grid resolu-tion by a factor of 4 increases the numericalcelerity to 88% of the linear theory value.
Using two-and-a-half equation turbulent clo-sures to characterize vertical mixing slightlyreduces the sharpness of the solution, but doesnot substantially change the celerity of the adjust-ment, and does not destroy positivity. All GLSclosures represented in ELCIRC_v5.01 providenearly indistinguishable results, shown in Fig. 10bfor k2k‘: Results for MY25 (not shown) aresimilar. This similarity should not be extrapolatedto more complex problems, as shown elsewhere(Baptista et al., 2004).
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UNCOR
Fig. 10. Adjustment under gravity of two fluids of different densities,
in a 64 km� 20 km� 20m domain, with Dx=Dy=500m, Dz=1m,
Background temperature was uniformly set at 4 1C. Initial salinity w
allowing for a sharp but continuous transition between 6.25 and 0 psu
after 12 h, in the form of isolines every 0.5 psu, are shown for (a) inv
TED PROOF
vertical mixing, stratification and Coriolis. Aparticularly demanding benchmark involves thesimulation of the plume resulting from a fresh-water discharge in a quiescent ocean, under strongCoriolis (e.g., at a mid-latitude). While noanalytical solution exists, key features of theexpected behavior include anti-cyclonic turningof the plume outside the river mouth, developmentof a narrow coastal jet in the direction of Kelvinwaves, very limited penetration on the opposite(‘‘upstream’’) direction, and surface trapping ofthe plume.We consider here the same forcings, physics and
computational domain used by Garcia-Berdeal etal. (2002) to solve this benchmark with ECOM3D,a POM derivative. The settings are loosely inspiredby typical winter conditions for the ColumbiaRiver plume. The domain consists of a deeprectangular ‘‘ocean’’ basin (140 km� 400 km),with a linearly sloping bottom (20m depth at thecoast and 300m at the offshore boundary). Joiningthe ‘‘ocean’’ at 120 km north of the southboundary is an estuary of size 10 km� 4 km witha constant depth of 20m. At the upstream riverboundary, freshwater is steadily discharged intothe top 10m of the vertical column at a constantrate of 7000m3 s�1. The south, west and northsides of the ‘‘ocean’’ are open. Coriolis is set atf=10�4 s�1. The simulation is cold started with a1-day ramp-up. Vertical mixing is represented withMY25, with background viscosity and diffusivityof 10�6m2 s�1. We also adopted the horizontal
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initially separated by a vertical ‘‘wall’’. Solutions were obtained
and Dt=300 s. No bottom friction or Coriolis were applied.
as imposed as S ¼ ð6:25=2Þ½1� tanh½ðx � 32; 000Þ=1000��; ; thus, respectively, to the left and right of the initial ‘‘wall’’. Results
iscid laminar conditions, and (b) for a k2k‘ closure.
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discretization of Garcia-Berdeal et al. (2002), witha quadrangular grid with Dx=1.5 km andDy=2km. However, the differences betweenECOM3D and ELCIRC require different strate-gies for the vertical discretization: rather than s-coordinates, we used 43 z-levels with Dz varyingfrom 0.5 to 117m. We used Dt=5min.
Surface salinities at the end of a 14-daysimulation are depicted in Fig. 11, and show avery good qualitative match both with expectedkey features and with the results of Garcia-Berdealet al. (2002). This suggests that ELCIRC_v5.01has the potential to represent, individually and inoverall balance, the relevant physical processesinvolved in the description of complex plumebehavior.
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Fig. 11. Idealized plume under mid-latitude, Northern hemi-
sphere Coriolis. Results are shown after 14 days, in the form of
isolines of salinity at 4 psu increments. Simulations represent
expected plume characteristics: anti-cyclonic turning outside the
river mouth, development of a narrow coastal jet in the
northern direction, very limited penetration of freshwater
southwards, and (not shown) surface trapping.
UNCORREC5. Concluding remarks
This paper presents the formulation and basicskill assessment of ELCIRC, an open-sourcenumerical model for cross-scale simulation ofriver-to-ocean 3D circulation. A member byconceptual affinity, although not by implementa-tion, of the class of finite volume Eulerian–La-grangian models introduced by Casulli and Cheng(1992) and Casulli and Zanolli (1998), ELCIRCfills a void by opening such class of models tobroad community use and feedback.
ELCIRC is a low-order model, which requireshighly resolved horizontal and vertical grids, butenables large time steps (with Courant numbers inexcess of unity) and appears capable of addressinga large variety of riverine and marine processes.The low-order and Eulerian–lagrangian nature ofsolution strategies in ELCIRC are in dramaticcontrast with those of community models such asADCIRC, QUODDY, POM, ROMS and SEOM,thus providing a truly distinctive alternative.
The Columbia River system, with its diversechallenges and tightly coupled scales and pro-cesses, provides an example of an applicationextremely well suited to a model like ELCIRC.Indeed, ELCIRC is now at the core of thecirculation modeling system that we have beendeveloping for the Columbia River, and supportssemi-operational generation of multi-year simula-
tions and routine daily forecasts of 3D baroclinicestuarine and plume circulation (Baptista et al.,2004).
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However, ELCIRC is still an emerging model.On-going developments and other desirable effortsinclude formulation and algorithm alternatives,MPI-based parallelization in distributed-memorycomputer clusters, and extension to ecologicalprocesses. Priority formulation and algorithmalternatives include a non-hydrostatic formula-tion; a fully conservative, higher-order solution ofscalar transports; and vertically stretched coordi-nates.
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UNCORREC
Acknowledgements
The development and testing of ELCIRC hasgreatly benefited from the contributions of manycolleagues. Through both his published work andinformal conversations with the second author inworkshops in Denver and Delft, Prof. VincenzoCasulli provided the conceptual foundation onwhich we built ELCIRC. Within the CORIE team,we are particularly grateful to Dr. Mike Zulauf fordeveloping air–water exchange modules (see Ap-pendix A); Dr. Arun Chawla for extended discus-sions on formulation and simulation quality; andMr. Paul Turner for the creation of supportingpre- and post-processing tools. We are also grate-ful to Dr. Edmundo Casillas, of the NationalOceanic and Atmospheric Administration(NOAA), for providing the driving applicationand partially facilitating the funded developmentof ELCIRC. ELCIRC has been developed througha combination projects funded by NOAA and theBonneville Power Administration (NA17FE1486,NA17FE1026, NA87FE0405), the US Fish andWildlife Service (133101J104), the Office of NavalResearch (N00014-00-1-0301, N00014-99-1-0051),and the National Science Foundation (ACI-0121475). Finally, we would like to thank earlybeta-testers of ELCIRC, who include CCALMRstudents Sergey Frolov, Michela Burla and Mar-ina Vantrease; Dr. Mike Foreman (Institute ofOcean Sciences, Canada); and Drs. AnabelaOliveira and Andre Fortunato (Laboratorio Na-cional de Engenharia Civil, Portugal). SAR images(Figs. 1a, c) were purchased by Dr. Todd Sanders,under NOAA NESDIS Grant NA16EC2450.
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Appendix A: Surface fluxes of heat and momentum
Atmospheric boundary conditions are requiredto model the exchange of heat and momentumbetween the atmosphere and the surface. Wedescribe here the representation of exchangesadopted in ELCIRC when results from weathermodels are available as forcings. An alternative,more simplified, approach to representing momen-tum exchanges is described in the text (Section2.3.1).The total heat transfer across the air–water
interface (into the water) is commonly described as
Htot #¼ ð1� AÞRs # þ RIR # �RIR "ð Þ � S � E;
(56)
where Htot # is the net downwards heat flux at theair–water interface, A is the albedo of the surface,Rs # is the downwelling solar radiative flux atsurface, RIR # and RIR " are the down/upwellinginfrared radiative fluxes a the surface, S is theturbulent flux of sensible heat (upwelling), and E isthe turbulent flux of latent heat (upwelling).As stated in Section 2.3.3, the non-solar heat
fluxes
H�tot # ¼ RIR # �RIR "ð Þ � S � E at z ¼ HR þ Z
(57)
are in ELCIRC applied as a surface boundarycondition, Eq. (16), to the heat transport equation,Eq. (6). This is appropriate because both theinfrared and turbulent fluxes essentially act at thesurface of the water. Conversely, solar radiation ispenetrative. The attenuation of solar radiation actsas a heat source within the water column, and isthus better expressed directly in Eq. (6), with
_Q ¼@R�
s
@z: (58)
The vertical profile of solar radiation is attenu-ated as in Paulson and Simpson (1977), given apredefined water type (Jerlov, 1968):
R�s ðdsÞ ¼ ð1� AÞRs # <e�dS=d1 þ ð1�<Þe�dS=d2
h i;
(59)
where ds is the water depth (defined in Eq. (31)),
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RREC
and d1; d2; and < are constants characterizing theturbidity of the chosen water type.
The downwelling radiative fluxes in Eqs. (57)and (58) are common output products of numer-ical weather prediction models, and the upwellinginfrared radiation may be approximated as theblackbody radiative flux from the water’s surface,
RIR "¼2 sT4sfc; (60)
where 2 is the emissivity ð� 1Þ; and s is theStefan–Boltzmann constant.
The turbulent fluxes of heat ðS;EÞ and momen-tum (t; Eq. (9)) are parameterized using the bulkaerodynamic formulation of Zeng et al. (1998).This parameterization has specifically been de-signed for improved accuracy in high-wind re-gimes, and takes into account surface layerstability, free convection, and variable roughnesslengths:
S ¼ �racpau�T�; E ¼ �raLeu�q�; t ¼ rau2�;
(61)
where T�; u�; and q� are scaling parameters for airtemperature, air velocity, and specific humidity; ra
is the surface air density, cpa is the specific heat ofair, and Le is the latent heat of vaporization. Thescaling parameters are defined using the dimen-sionless flux–gradient relations of Monin–Obu-khov similarity theory, and must be solved foriteratively. They depend upon near-surface tem-peratures or water and air, wind speed, andspecific humidity, as well as surface atmosphericpressure (all of which may also be obtained fromnumerical weather predictions).
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UNCOReferences
Baptista, A.M., 1987. Solution of advection-dominated trans-
port by Eulerian–Lagrangian methods using the backwards
methods of characteristics. Civil Engineering. Cambridge,
MA, USA, Ph.D. Thesis, Massachusetts Institute of
Technology, 260pp.
Baptista, A.M., 2002. Environmental Observation and Fore-
casting Systems. In: Meyers, R.A. (Ed.), Encyclopedia of
Physical Science and Technology, Vol. 5. Academic Press,
New York, p. 16.
Baptista, A.M., Adams, E.E., Stolzenbach, K.D., 1984.
Eulerian–Lagrangian analysis of Pollutant Transport in
TED PROOF
Shallow Water. MIT R. M. Parsons Laboratory, Cam-
bridge, MA, USA.
Baptista, A. M., Wilkin, M., Pearson, P., Turner, P.,
McCandlish, C., Barrett, P., Das, S., Sommerfield, W., Qi,
M., Nangia, N., Jay, D., Long, D., Pu, C., Hunt, J., Yang,
Z., Myers, E., Darland, J., Farrenkopf, A., 1998. Towards a
multi-purpose forecast system for the Columbia River
Estuary. Ocean Community Conference ‘98, Baltimore,
MD, USA.
Baptista, A.M., Wilkin, M., Pearson, P., Turner, C., McCandl-
ish, C., Barrett, P., 1999. Coastal and estuarine forecast
systems: a multi-pupose infrastructure for the Columbia
River. Earth Sys. Mon./NOAA 9(3).
Baptista, A. M., Y. Zhang, A. Chawla, E. Myers, M. Zulauf,
2004. A cross-scale model for 3D baroclinic circulation in
estuary–plume–shelf systems: II. Application to the Colum-
bia River. Continental Shelf Research, in preparation.
Blumberg, A.F., Mellor, G.L., 1987. A description of a three-
dimensional coastal ocean circulation model. In: Heaps, N.
(Ed.), Three-Dimensional Coastal Ocean Models, Vol. 4.
AGU, Washington, DC, USA, pp. 1–16.
Canuto, V.M., Howard, A., Cheng, Y., Dubovikov, M.S.,
2001. Ocean turbulence I: one-point closure model. Mo-
mentum and heat vertical diffusivities. Journal of Physical
Oceanography 31, 1413–1426.
Casulli, V., Cattani, E., 1994. Stability, accuracy and efficiency
of a semi-implicit method for 3D shallow water flow.
Computers & Mathematics with Applications 27, 99–112.
Casulli, V., Cheng, R.T., 1992. Semi-implicit finite difference
methods for three-dimensional shallow water flow. Interna-
tional Journal of Numerical Methods in Fluids 15, 629–648.
Casulli, V., Zanolli, P., 1998. A three-dimensional semi-implicit
algorithm for environmental flows on unstructured grids.
Institute for Computational Fluid Dynamics Conference on
Numerical Methods for Fluid Dynamics VI.
CCALMR, 2003. ELCIRC. WWW Page. http://www.ccalm-
r.ogi.edu/CORIE/modeling/elcirc, Center for Coastal and
Land-Margin Research (CCALMR), Oregon Health &
Science University, OR, USA.
Flather, R.A., 1987. A tidal model of Northeast Pacific.
Atmosphere–ocean 25 (1), 22–45.
Foreman, M.G.G., Henry, R.F., Walters, R.A., Ballantyne,
V.A., 1993. A finite element model for tides and resonance
along the north coast of British Columbia. Journal of
Geophysical. Research 98 (C2), 2509–2531.
Galperin, B., Kantha, L.H., Hassid, S., Rosati, A., 1988. A
quasi-equilibrium turbulent energy model for geophysical
flows. Journal of Atmospheric Science 45, 55–62.
Garcia-Berdeal, I., Hickey, B.M., Kawase, M., 2002. Influence
of wind stress and ambient flow on a high discharge river
plume. Journal of Geophysical. Research 107 (C9), 3130.
Haidvogel, D.B., Arango, H.G., Hedstrom, K., Beckmann, A.,
Malanotte-Rizzoli, P., Shchepetkin, A.F., 2000. Model
evaluation experiments in the North Atlantic Basin:
simulations in nonlinear terrain-following coordinates.
Dynamics of Atmosphers and Oceans 32, 239–281.
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
51
53
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
85
87
89
91
93
ARTICLE IN PRESS
CSR : 1149
Y. Zhang et al. / Continental Shelf Research ] (]]]]) ]]]–]]]28
UNCORREC
Iskandarani, M., Haidvogel, D.B., Levin, J., 2003. A three-
dimensional spectral element model for the solution of the
hydrostatic primitive equations. Journal of Computational
Physics 186 (2), 397–425.
Jerlov, N.G., 1968. Optical Oceanography. Elsevier, Amster-
dam.
Kantha, L.H., Clayson, C.A., 1994. An improved mixed layer
model for geophysical applications. Journal of Geophyscial
Research 99 (25), 235–266.
Lehfeldt, R., Bloss, S., 1988. Algebraic turbulence model for
stratified tidal flows. In: Dronkers, J., Leussen, W.v. (Eds.),
Physical processes in estuaries. Springer, New York.
Luettich, R.A., Westerink, J.J., Scheffner, N. W., 1991.
ADCIRC: an advanced three-dimensional circulation model
for shelves, coasts and estuaries, Coast. Engrg. Res. Ct., US
Army Engs. Wtrways. Experiment Station, Vicksburg, MS.
Luettich, R.A., Muccino, J.C., Foreman, M.G.G., 2002.
Considerations in the calculation of vertical velocity in
three-dimensional circulation models. Journal of Atmo-
spheric and Oceanic Technology 19 (12), 2063–2076.
Lynch, D.R., Gray, W.G., 1978. Analytic solutions for
computer flow model testing. ASCE Journal of the
Hydraulics Division 104 (HY10), 1409–1428.
Lynch, D.R., Ip, J.T., Naimie, C.E., Werner, F.E., 1996.
Comprehensive coastal circulation model with application
to the Gulf of Maine. Continental Shelf Research 16 (7),
875–906.
Mellor, G.L., Yamada, T., 1982. Development of a turbulence
closure model for geophysical fluid problems. Reviews in
Geophys 20, 851–875.
Millero, F.J., Poisson, A., 1981. International one-atmosphere
equation of state of sea-water. Deep-Sea Research 28A,
625–629.
Munk, W.H., Anderson, E.R., 1948. Notes on a theory of the
thermocline. Journal of Marine Research 7, 276–295.
Myers, E., Aikman, F., 2003. A forecast circulation model of
the St. Johns River, Florida. 8th Estuarine and Coastal
Modeling Conference, Monterey Bay, CA, USA.
Oey, L.Y., Mellor, G.L., 1993. Subtidal variability of estuarine
outflow, plume, and coastal current: a model study. Journal
of Physical Oceanography 23, 164–176.
Oliveira, A., Baptista, A.M., 1995. A comparison of integration
and interpolation Eulerian-Lagrangian methods. Interna-
tional Journal of Numerical Methods in Fluids 21, 183–204.
Oliveira, A., Baptista, A.M., 1998. On the role of tracking on
Eulerian-Lagrangian solutions of the transport equation.
Advances in Water Resources 21, 539–554.
Oliveira, A., Fortunato, A., Baptista, A.M., 2000. Mass
Balance in Eulerian-Lagrangian Transport Simulations in
Estuaries. ASCE Journal of Hydraulic Engineering 126 (8),
605–614.
Pacanowski, R.C., Philander, S.G.H., 1981. Parameterization
of vertical mixing in numerical models of tropical oceans.
Journal of Physical Oceanography 11, 1443–1451.
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Paulson, C.A., Simpson, J.J., 1977. Irradiance measurements in
the upper ocean. Journal of Physical Oceanography 7,
952–956.
Pinto, L.L., Oliveira, A., Fortunato, A. B., Baptista, A. M.,
2003. Analysis of the stratification in the Guadiana estuary.
8th Estuarine and Coastal Modeling Conference, Monterey
Bay, CA.
Pond, S., Pickard, G.L., 1998. Introductory Dynamical
Oceanography. Butterworth-Heinmann, Stoneham, MA.
Reid, R.O., 1990. Waterlevel changes. Handbook of Coastal
and Ocean Engineering. Gulf Publishing, Houston, Tx.
Robinson, C.L.K., Morrison, J., Foreman, M.G.G., 2004.
Oceanographic connectivity among marine protected areas
on Canada’s northern Pacific coast. Marine Ecology
Progress Series (submitted).
Rodi, W., 1984. Turbulence models and their applications in
hydraulics: a state of the art review. Delft, The Netherlands,
International Association for Hydraulics Research.
Song, Y., Haidvogel, D., 1994. A semi-implicit ocean circula-
tion model using a generalized topography-following
coordinate system. Journal of Computatinal Physics 115,
228–244.
Turner, P.J., Baptista, A.M., 1991. User’s manual: software for
semi-automatic generation of two-dimensional finite ele-
ment grids, OGI-CCALMR Software Documentation
Series SDS2–91/2.
Umlauf, L., Burchard, H., 2003. A generic length-scale
equation for geophysical turbulence models. Journal of
Marine Research 6 (12), 235–265.
Vaz, R.A., Simpson, J.H., 1994. Turbulence closure modeling
of estuarine stratification. Journal of Geophysical Research
99 (C8), 16143–16160.
Wang, D.-P., 1984. Mutual intrusion of a gravity current and
density front formation. Journal of Physical Oceanogr 14,
1191–1199.
Warner, J.C., Sherwood, C.R., Arango, H.G., Signell, R.P.,
2004. Performance of four turbulence closure models
implemented using a generic length scale method. Ocean
Modeling (in press).
Wilcox, D.C., 1998. Reassessment of scale determining
equation for advance turbulence models. AIAA Journal
26, 1299–1310.
Wood, T.M., Baptista, A.M., Kuwabara, J.S., Flegal, A.R.,
1995. Diagnostic modeling of trace metal partitioning in
south San Francisco Bay. Limnology in Oceanography 40
(2), 345–358.
Zeng, X., Zhao, M., Dickinson, R.E., 1998. Intercomparison of
bulk aerodynamic algorithms for the computation of sea
surface fluxes using TOGA COARE and TAO data. Journal
of Climatology 11, 2628–2644.
Zhang, Y., Baptista, A.M., 2000. An efficient adaptive editor
for unstructured grids. 7th International Conference on
Numerical Grid Generation in Computational Field Simu-
lations, Whistler, BC, Canada.
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