J. Great Lakes Res. 30 (Supplement 1):41–54Internat. Assoc. Great Lakes Res., 2004

41

A Non-orthogonal Primitive Equation Coastal OceanCirculation Model: Application to Lake Superior

Changsheng Chen1,*, Jianrong Zhu2, Lianyuan Zheng3, Elise Ralph4, and Judith Wells Budd5

1School of Marine Science and TechnologyUniversity of Massachusetts-Dartmouth

New Bedford, Massachusetts 02742

2State Key Laboratory of Estuarine and Coastal ResearchEast China Normal UniversityShanghai 200062, P. R. China

3College of Marine ScienceUniversity of South Florida

St. Petersburg, Florida 33701

4The Large Lakes ObservatoryUniversity of Minnesota

Duluth, Minnesota 55812

5Department of Geological Engineering and SciencesMichigan Technological University

Houghton, Michigan 49931

ABSTRACT. A non-orthogonal coordinate primitive equation model has been developed for the studyof the Keweenaw Current in Lake Superior. This model provides a more accurate fitting of the coastline.A comparison with a curvilinear orthogonal model shows that the non-orthogonal transformation modelprovided a better simulation of the current jet in the near-shore region. Accurate fitting of both bathyme-try and irregular coastlines plays an essential role in capturing the magnitude of the Keweenaw Currentand cross-shelf structure of the thermal bar near the coast. The formation of the Keweenaw Current andthermal front was directly driven by a westerly or southwesterly wind and seasonal development of strati-fication over steep bottom topography. Under a condition with accurate fitting of steep bathymetry, fail-ure to resolve the irregular geometry of the coastline can result in an underestimation of the magnitude ofthe Keweenaw Current by about 20 cm/s.

INDEX WORDS: Numerical modeling, thermal bar, Keweenaw Current, Lake Superior, numericalmethods.

*Corresponding author. E-mail: c1chen@umassd.edu

INTRODUCTION

Lake Superior, one of the largest lakes in theworld, has an east-west length of about 600 km anda south-north width of 250 km. It has a coastline ofabout 5,000 km, with a narrow “continental shelf”of about 10–15 km around the coast. The averageslope of the shelf is about 0.01 on the northerncoast, about 0.05 on the western and eastern coasts,

and 0.035 on the southern coast. The steepest slopeof the shelf, exceeding 0.05, occurrs along the Ke-weenaw Peninsula (Fig. 1). In the interior, the meandepth is about 150 m, but the maximum depth isgreater than 400 m.

The circulation in Lake Superior is driven mainlyby winds and buoyancy (Jacobs 1974, Bennett1978, Chen et al. 2001, Zhu et al. 2001). Currentsvary significantly with seasons. They are weak andreverse intermittently in spring and fall due to fre-quent changes of the wind’s direction under the

42 Chen et al.

condition of weak stratification. During summer,currents are stronger and relatively stable, as aresult of the dominance of the northeastward windplus the formation and intensification of coastalthermal front (Hubbard and Spain 1973, Spain et al.1976). A northeastward coastal current (called theKeweenaw Current) is a nearly permanent featurealong the Keweenaw coast in summer, forming amajor characteristic of the circulation in Lake Supe-rior (Ragotzkie 1966, Smith and Ragotzkie 1977,Niebeauer et al. 1977, Phillips 1978, Viekman andWimbush 1993). The Keweenaw Current is con-fined within a narrow near-shore band of order 5-km in width. It is affected significantly by the innershelf physical processes associated with wind mix-ing, and wind-induced convergence (downwelling)and divergence (upwelling) near the coast.

One of the objectives of the Keweenaw Interdis-ciplinary Transport Experiment in Superior Lake(KITES) is to determine the effects of physical,chemical, and biological processes that arise fromthe formation, evolution, and perturbation of thethermal bar and Keweenaw Current on the distinctnearshore and offshore ecosystems in Lake Supe-rior. Since these processes are nonlinearly coupledwith each other, it is difficult to examine, verify,

and understand the ecosystem dynamics merelythrough observations taken either at a given time orat a few monitoring stations. Developing a vali-dated full three-dimensional, prognostic numericalmodel will provide us with a scientific tool toexamine these complex environmental problems.

In the last 30 years, many efforts have been madeon developing the Lake Superior model. Lam(1978) was a first person who developed a four-layer model. Driven by realistic wind and buoyancyforcing, his model did produce a current jet alongthe Keweenaw coast, but the model-predicted speedwas only about 5 to 7 cm s–1, that is, 5 to 10 timessmaller than that observed by Niebauer et al.(1977). The cases for this underestimation werebelieved due to (1) the low cross-shelf model reso-lution, (2) the lack of surface heating, and (3) thefailure to resolve irregular coastal geometry (Chenet al. 2001; Zhu et al. 2001). A uniform horizontalresolution of 10 km was used in Lam’s (1978)model experiment. This numerical resolution waslarger than the general cross-shelf scale of theKeweenaw Current observed in summer. The modelcomputational domain was configured with struc-tured rectangular grid meshes. These meshes poorlymatched the shape of the coastline along theKeweenaw Peninsula. The step-like boundary con-structed from these rectangular meshes causes arti-ficial drag force to slow down the along-coast cur-rent.

Chen et al. (2001) developed a so-called non-orthogonal coordinate transformation oceanicmodel for Lake Superior. The non-orthogonal coor-dinate transformation allows us to make numericalmeshes more flexible in shapes by removing therequirement of a strict orthogonal (90°) condition ofmesh angles. The model with the non-orthogonalcoordinate transformation provides a better fittingof irregular coastline, which is believed to play acritical role in the simulation of the near-shore cur-rent jet. This new model has successfully simulatedthe currents and thermal front (in the Great Lakescommunity, it is called “thermal bar”) observedalong the Keweenaw coast in Lake Superior (Chenet al. 2001, Zhu et al. 2001). The model-predictedwater transport has been used to guide the processstudies of the impact of physical processes on thetemporal and spatial distributions of the biologicalvariables observed during the KITE field measure-ments.

One of the critical fundamental technical issueswas raised on the non-orthogonal coordinationtransformation model: Do we really need such a

FIG. 1. Bathymetry of Lake Superior (a) andmoored buoy sites (b). Solid circles are the loca-tions of moored current meter labeled 1, 2 and 3.The unit of the bathymetry is meters.

A Lake Superior Circulation Model 43

model to be successful in simulating the water cur-rent and transport in Lake Superior? The curvilinearorthogonal coordinate transformation model, whichrequires all angles of individual numerical mesh tosatisfy 90°, has been widely used in the GreatLakes community. These models provide a moder-ate fit for the irregular coastal boundary. However,because this method requires a strict orthogonal(90°) condition, a grid generation program using afast Poisson solver of the Laplace equation makesreaching a convergence solution difficult and alsofails to fit an irregular coastline characterized withcomplex geometric boundary (like one shown inKeweenaw Peninsula). Applying this transforma-tion to the coast with a concave bend usually resultsin a poor resolution near the coast.

The limitation of a curvilinear orthogonal coordi-nate transformation model in application to coastalregions has been recognized for many years in thecoastal ocean community. The successful applica-tion of the non-orthogonal coordinate transforma-tion model to Lake Superior indirectly implies theimportance of the geometric fitting in that area.Since no quantitative comparisons between orthog-onal and non-orthogonal coordinate transformationmodels have been made regarding the realistic sim-ulation, the accuracy of the non-orthogonal coordi-nate transformation oceanic model is still not vali-dated.

The objective of this study is to provide a quanti-tative validation of the importance of coastal geo-metric fitting in simulating the Keweenaw Currentin Lake Superior. To achieve this goal, we re-ranthe simulation using both curvilinear orthogonaland non-orthogonal coordinate transformation ver-sions of ECOM-si (Blumberg 1994) and comparedthe model results with the observed currents atthree buoy sites off Eagle Harbor along theKeweenaw coast. A sensitivity study clearly showsthat the non-orthogonal transformation model pro-vides a better simulation of the current in the near-shore region. Accurate fitting of bathymetry andirregular coastlines plays an essential role in captur-ing the magnitude of the Keweenaw Current andcross-shelf structure of the thermal front near thecoast. The formation of the Keweenaw Current andthermal front was directly driven by the westerly orsouthwesterly wind and seasonal development ofstratification over steep bottom topography. Undera condition with accurate fitting of steep bathyme-try, failure to resolve the irregular geometry of thecoastline can result in an underestimation of the

magnitude of the Keweenaw Current by about 20cm/s.

The remaining sections are organized as follows.The non-orthogonal model formuilation sectiondescribes the primitive equation of the coastalmodel with a non-orthogonal coordinate transfor-mation. The numerical computational method sec-tion represents the numerical methods used to solvethe model. The model application and validationsection shows examples of the application of thismodel to Lake Superior. Finally, the conclusion isgiven in the last section.

THE NON-ORTHOGONALMODEL FORMULATION

The Primitive Equations

The governing equations of ocean circulation andwater masses consist of momentum, continuity,temperature, salinity, and density equations as fol-lows:

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

− =

− ∂∂

− ∂∂

+ ∂∂

u

tu

u

xv

u

yw

u

zfv

gx

P

x zKm

ζ ∂∂∂

+u

zFu

(1)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

+ =

− ∂∂

− ∂∂

+ ∂∂

v

tu

v

xv

v

yw

v

zfu

gy

P

y zKm

ζ ∂∂∂

+v

zFv

(2)

∂∂

+ ∂∂

+ ∂∂

=u

x

v

y

w

z0 (3)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

= ∂∂

∂∂

+ ∂∂

θ θ θ θ θt

ux

vy

wz z

Kz

SWh zz

F+ θ (4)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

= ∂∂

∂∂

+s

tu

s

xv

s

yw

s

z zK

s

zFh s (5)

ρtotal = ρtotal (θ,s ) (6)

where x, y, and z are the east, north, and vertical axesof the Cartesian coordinate; u, v, and w the x, y, zvelocity components; θ the potential temperature; sthe salinity; ζ the sea surface elevation; P the baro-clinic pressure defined as 1/ρo ∫

0

zρgdz; f the Coriolis

parameter; g the gravitational acceleration; Km thevertical eddy viscosity coefficient; Kh the thermalvertical eddy diffusion coefficient, and SW is the

44 Chen et al.

short-wave solar radiation that is a function of z andt. Fu, Fv, Fθ, and Fs represent the horizontal momen-tum, thermal, and salt diffusion terms. ρ and are theperturbation and reference density, which satisfy

ρtotal = ρ + ρo. (7)

Fu, Fv, Fθ, and Fs are calculated by Smagorinsky’sformula (Smagorinsky 1963) in which the horizon-tal diffusion is directly proportional to the productof horizontal grid sizes. Km and Kh are calculatedusing the Mellor and Yamada (Mellor and Yamada1974, 1982) level 2.5 turbulent closure scheme,with the modified stability function by Galperin etal. (1988). This turbulent-closure model is designedto simulate boundary layer physics through theinclusion of (1) shear and buoyancy production ofturbulent kinetic energy, (2) dissipation of turbulentkinetic energy, (3) vertical diffusion and the timederivative of the turbulent kinetic energy and turbu-lent macroscale, and (4) the advection of the turbu-lent kinetic energy.

The Non-Orthogonal Coordinate System

Let us define a new coordinate transformation as

ξ = ξ(x, y), η = η(x, y) (8)

Taking the differentiation of Eq. (8) with respect tox and y, respectively, yields the interchange rela-tions as

where J is the Jacobian function in the form of

J = xξ yη – xηyξ (10)

and the subscript symbols (ξ and η) indicate deriva-tives. The metric factors h1 and h2 of the coordinatetransformation are defined as

(Fig. 2) and the interchange relation of unit vectorsbetween non-orthogonal (ξ,η) and orthogonal (x, y)coordinates is given by

ξ ξ η ηη η ξ ξx y x y

y

J

x

J

y

J

x

J= = − = − =, , , (9)

eh

y i x j eh

y i x jξ η η ξ ξ ξ= −( ) = − +(1 1

2 1

, )) , (13)

uh

Jx u y v v

h

Jx u y v1

21

1= + = +( ),� � � � � � ( ).ξ ξ η η (14)

∂∂

+ ∂∂

+ ∂∂

=JU JV Jw

z

ˆ ˆ

ξ η0 (17)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

= ∂∂

∂∂

+s

tU

sV

sw

s

z zK

zFh s

ˆ ˆξ η

θ(19)

ˆ , ˆUJ

h uh

hv V

Jh v

h= −

= −1 12 1

3

11 1 1

3

hhu

21

(20)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

= ∂∂

∂∂

+ ∂θ θξ

θη

θ θt

U V wz z

Kz

Sh

ˆ ˆ WW

zF

∂+ θ (18)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

−

∂∂

u

tU

uV

uw

u

z

h V

J

vJ

h

1 1 1 1 2

11

ˆ ˆˆ

ξ η

ξ

− ∂∂

+

−

∂∂

uJ

hJf

h

Ju v

12

21 1

η

ξhh

h h

h

J

P

zK

u

zF

om

3

1 2

2 1

= − ∂∂

+ ∂∂

∂∂

+ρ ξ xx

(15)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

+

∂∂

v

tU

vV

vw

v

z

h U

J

vJ

h

1 1 1 1 1

11

ˆ ˆˆ

ξ η

ξ

− ∂∂

+

−

∂∂

uJ

hJf

h

Ju v

12

11 1

η

ηhh

h h

h

J

P

zK

v

zF

om

3

1 2

1 1

= − ∂∂

+ ∂∂

∂∂

+ρ η yy

(16)

h x y h x y12 2

22 2= + = +ξ ξ η η, . (11)

ei j

ei jx y

x y

x yξ η

ξ ξ

ξ ξ

η η

η=

+

+=

+2 2

,xx y2 2+ η

, (12)

The unit vectors for the coordinate (ξ,η) aredefined as

( ,� � � )!e ehξ

Therefore, the ξ and η components of the velocity(defined as u1, v1) can be expressed in the form of

Using this transformation, we can re-write themomentum, continuity, temperature and salinityequations aswhere

where

A Lake Superior Circulation Model 45

and

h3 = yξyη + x ξxη (21)

are the cross-metric factors caused by the non-orthogonal transformation. The non-orthogonal

coordinate transformation used here is completelydifferent from the one introduced in CH3D orCH3D-WES by Sheng (1986) and Chapman et al.(1996). A detailed discussion on the difference isgiven in Appendix.

FIG. 2. Illustration of the non-orthogonal coordinate system used in the model.

46 Chen et al.

The σσ-Coordinate Transformation System

To simplify the numerical calculation over anirregular bottom slope, a vertically stretched σ-coordinate transformation is used in the numericalcomputation. This transformation is defined as

where H(x,y) is the mean water depth and D = H +ζ. The vertical coordinate σ varies from −1 at z = −H to 0 at z = ζ. In this system, eqs. (15)–(19) canbe rewritten with the form of the transport as

where

σ ζζ

= −+

z

H(22)

ω σξ η

σ ζ ζξ

= − ∂∂

+ ∂∂

−

+ ∂∂

+ ∂∂

+

w UD

VD

tU

ˆ ˆ

( ) ˆ ˆ1 VV∂∂

ζη

(28)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

−DJu

t

DJUu DJVu J u

Dh V v

1 1 1 1

2 1

ˆ ˆ

ˆ

ξ ηωσ

∂∂∂

− ∂∂

+

−ξ η

J

hu

J

hJf Dh

11

22uu v

h

h h

h gDgh D D

o

1 13

1 2

22

∂∂

= − ∂∂

+ ∂∂

∂

ξ

ζξ ρ ξ

σ ρρσ

σρ ξ

ρ σσ σ

σ

σ

∂− ∂

∂

+ ∂∂

∂∂

∫

∫

dgh D

dD

KJu

o

m

220

011

+ DJFx

(23)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

+DJv

t

DJUv DJVv J v

Dh U v

1 1 1 1

1 1

ˆ ˆ

ˆ

ξ ηωσ

∂∂∂

− ∂∂

+

−ξ η

J

hu

J

hJf Dh

11

21uu v

h

h h

h gDgh D D

o

1 13

1 2

11

∂∂

= − ∂∂

+ ∂∂

∂

η

ζη ρ η

σ ρρσ

σρ η

ρ σσ σ

σ

σ

∂− ∂

∂

+ ∂∂

∂∂

∫

∫

dgh D

dD

KJv

o

m

120

011

+ DJFy

(24)

∂∂

+ ∂∂

+ ∂∂

+ ∂

∂=ζ

ξ ηωσt J

DJU DJV1

0( ˆ ) ( ˆ ) (25)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

=

∂∂

∂∂

JDs

t

JDUs JDVs J s

DK

Jsh

ˆ ˆ

ξ ηωσ

σ1

σσ

+ DJFs

(27)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

=

∂∂

∂∂

JD

t

JDU JDV J

DK

Jh

θ θξ

θη

ωθσ

σθ

ˆ ˆ

1σσ σ θ

+ ∂∂

+SWDJF

(26)

Boundary and Initial Conditions

Eqs. (23)–(28) and Eq. (6) are solved prognosti-cally as initial value problems of oceanic motion.The initial condition of the velocity takes u1 = v1 =0. The boundary conditions are specified based onthe oceanic problems we will study. For example,for closed basins like the Great Lakes, the boundarycondition is specified as vn = 0 where vn is the nor-mal velocity component at the boundary. In thesemi-enclosed basin or continental shelf, the sea-surface elevation (amplitude and phase of tidal forc-ing) is specified at the open boundary. The surfaceboundary conditions include (1) wind stress, (2) netsurface heat flux and (3) precipitation/evaporation.Time variable river/dam and onshore intake/outfalldischarges also are included as lateral boundary con-ditions in the model for the study of the buoyancy-driven circulation caused by river discharges.

THE NUMERICALCOMPUTATIONAL METHOD

Eqs. (22)–(27) are very similar to the primitiveequations in the curvilinear orthogonal and σ-coor-dinate systems used in the POM or ECOM-siexcept for extra terms related to h3, Û and V̂. Thisnon-orthogonal coordinate transformation model isidentical to the ECOM-si as h3 = 0, and J = h1h2 inan orthogonal case. The numerical methods usedfor solving the ECOM-si are directly adopted tothis non-orthogonal coordinate transformationmodel with some modifications. To provide moreaccurate water, heat and salt transports, the nonlin-ear advection terms in momentum, temperature, andsalinity equations of the non-orthogonal coordinatemodel are calculated using the multidimensionalpositive definite advection transport algorithm(Smolarkiewicz 1984, Smolarkiewicz and Clark1986). A semi-implicit scheme is used for the timeintegration (Casulli 1990), in which the free surfacegradients in the momentum equations and transportterms in the continuity equations are treated implic-itly. In the original code of the ECOM-si, the solu-tion for the free surface elevation is achieved bysolving a linear, symmetric, 5-diagonal matrix sys-

A Lake Superior Circulation Model 47

tem. In the non-orthogonal coordinate transforma-tion model, however, the extra terms in the momen-tum equations result in a 9-diagonal matrix systemthat cannot be solved efficiently by preconditionedconjugate gradient methods. For this reason, wetreat all terms related to h3 using the values in theprevious time step, then the equations used to cal-culate the free surface elevation become a positivedefinite, symmetric, and 5-diagonal system as thesame as the ECOM-si (Blumberg and Mellor 1987).After this treatment, the non-orthogonal coordinatetransformation model can be solved numericallyusing the same semi-implicit approached used inthe ECOM-si. Vertical diffusion terms in bothmomentum and temperature/salinity equations alsoare computed implicitly. When the advections andhorizontal/vertical diffusions vanish in the momen-tum balance, the Euler forward scheme used in theECOM-si becomes numerically unstable (Mesingerand Arakawa 1976). A predicted-corrector scheme(Wang and Ikeda 1995) is adopted to compute theCoriolis terms in the momentum equations.

An Arakawa C-grid is used in the numerical inte-gration (Arakawa 1966). The grid generation pro-gram is developed by modifying Winslow’s methodfor the automatic generation of computationalmeshes (Winslow 1966, Brackbill and Saltzman1982). Winslow’s method requires us to solve anonlinear Poisson equation for generating a gridmapping from a regular numerical domain to anirregular-shape physical domain. This method isextended to adaptive mesh with varying zone sizesand orthogonality of grid lines (Brackbill and Saltz-man 1982). We adopt this method to create orthogo-nal grids off the coast or near the open boundaries.We then employ our own “Matlab” program tomodify grids near the coast to assume the shape ofquadrilaterals for fitting the coastlines. Thisapproach helps us avoid numerical instability prob-lems that may occur due to improper treatment ofopen boundary conditions in the non-orthogonalcoordinate transformation model.

THE MODEL APPLICATION ANDVALIDATION

To test our new non-orthogonal coordinate trans-formation oceanic circulation model, we applied itto Lake Superior for the study of coastal thermalfront and a near-shore current jet (Fig. 1). In thewestern shore of Lake Superior’s Keweenaw Penin-sula, the coastal current is characterized by a strongcoastal jet known as the Keweenaw Current (Har-

rington 1895, Hooper et al. 1973). The formation ofthis current is due to an intense temperature frontover steep bottom topography near the shore and anonshore Ekman transport caused by westerly orsouthwesterly winds (Smith and Ragotzkie 1970,Chen et al. 2001, and Zhu et al. 2001). Because thecross-shelf scale of the thermal front and current jetis less than 10 km over a sharp decrease of the bot-tom topography, a proper fitting of the coastline andbottom topographies becomes essential for a suc-cessful simulation of the Keweenaw Current. In ourmodel experiments, the irregular bathymetry is fit-ted by the σ-transformation coordinate system inthe vertical. Under sufficient horizontal resolution,this transformation can accurately resolve thedynamics related to the steep bottom topography.The non-orthogonal coordinate transformation inthe horizontal provides an accurate matching of theirregular coastline, which avoids a velocity errordue to an improper solid boundary condition.

To validate our non-orthogonal grid Lake Supe-rior model, we compared the simulation results ofthe near shore Keweenaw Current in southern LakeSuperior obtained from curvilinear orthogonal andnon-orthogonal transformation models. The valida-tion of these two models was based on the agree-ment with observations. The model domain for bothorthogonal and non-orthogonal cases is shown inFigure 3, which covers the entire lake with a highresolution of grids near the Keweenaw coast. Thehorizontal model grid cells were 123 (along-shelf)× 126 (cross-shelf ). Horizontal resolution wasabout 300 to 600 m in the cross-shelf direction and4 to 6 km in the along-shelf direction off theKeweenaw coast. The model grid cells were orthog-onal in the interior of the lake for both cases. In thenon-orthogonal case, the quadrilaterals were used tofit the irregular shape of the coastline (Fig. 4a),while in the orthogonal case, the coastline wastreated as a smoothed curvature line as shown inFigure 4b. To satisfy the restriction of the exactorthogonal requirement, the boundary cells in theorthogonal case reduced a certain level of the reso-lution as that shown in the non-orthogonal case. Inboth the orthogonal and non-orthogonal cases, the31 uniform σ-levels are used in the vertical, whichresulted in a vertical resolution of about 1 m nearthe coast and 10 m at the 300-m isobath in the inte-rior of the lake. The time step was 360 seconds.

The model was forced by observed winds andheat flux taken in July 1973 near Eagle Harbor. Thesurface wind stress was calculated using Large andPond’s formula with a neutral, steady-state drag

48 Chen et al.

coefficient (Large and Pond 1981). The heat fluxdata included both net surface heat flux and short-wave radiation. The initial stratification was set upusing observed temperature data taken regionallyover the lake in early July. Surface temperature wasabout 14°C at the coast and decreased to 5°C over across-shelf distance of 5 km. The large vertical tem-perature gradient only existed in the upper 60 m.

Figure 5 shows the comparisons between model-predicted and observed current speed at the near-surface at buoy stations 1–3 (see Fig. 2 for the loca-tions of these three buoys). The current data usedfor this comparison was provided by Joe Niebauerat the University of Wisconsin-Madison (Niebaueret al. 1977). The observational data were taken inJuly of 1973. Buoys 1 and 2 were deployed about0.7 km and 2.5 km off the coast near Eagle Harbor.The model-predicted currents calculated using thenon-orthogonal coordinate transformation modelwere in good agreement with observed currents atall three buoy stations, while the orthogonal coordi-nate transformation model significantly underesti-mated the magnitude of the current at near-shorebuoy stations 1 and 2. Because the original orthogo-nal and newly developed non-orthogonal versions

of the model have the exact same physics, the dif-ference found in these two cases is due to the accu-racy of the coastline fitting. This also explains whyno numerical simulations could successfully simu-late the Keweenaw Current in Lake Superior untilour efforts using this new non-orthogonal coordi-nate transformation model.

The new non-orthogonal coordinate transportmodel introduced here also provided a reasonablesimulation of the thermal front and Keweenaw Cur-rent measured during the KITES field programs.For example, the satellite-derived SST showed anarrow thermal front on 17 September 1999 (Fig. 6:upper-left). This front was reasonably captured bythe model simulation (Fig. 6: lower-left). TheADCP data received at site E3 showed two strongalong-shore currents during 9–11 September and13–15 September, respectively. The currents weak-ened with depth, having a vertical scale of about 60and a speed of 60 cm/s at a depth of 17 m (Fig. 6:upper-right). These along-shore currents were cap-tured up reasonably by the model (Fig. 6: lower-right). These comparisons suggest that this model isa robust tool for the study of the near-shore physi-cal process in Lake Superior.

FIG. 3. Non-orthogonal coordinate transformation model grids for Lake Superior.

A Lake Superior Circulation Model 49

CONCLUSION

A non-orthogonal coordinate primitive equationmodel has been developed for the study of theinner-shelf coastal oceanic circulation. A version ofthe non-orthogonal coordinate transformation usedin this model provides a more accurate fitting of thecoastline, which allows us to simulate the near-coastal physical processes such as tides, coastalfronts, and current jet. By removing the require-ment of a strict orthogonal condition, this modelprovides an easier and faster convergence for gridgeneration. The model is computed numericallyusing a finite difference method. The numericalprogram of the model has been written based on thesemi-implicit code of Blumberg and Mellor’s curvi-linear orthogonal coastal ocean circulation model(ECOM-si) with some modifications to allow for

inclusion of non-orthogonal transformation termsand correction of the calculation of the Coriolisterms. Therefore, users who are familiar with POMor ECOM-si can easily learn how to use this model.

The numerical program of this non-orthogonalgrid model was tested first by running it parallelwith ECOM-si under orthogonal grids for the studyof the formation of the thermal front and current jetalong the Keweenaw coast. Given the same hori-zontal resolution, the comparison between themodel-predicted currents obtained by the modelruns with orthogonal and non-orthogonal coordi-nate transformations shows that the non-orthogonalgrid model provided much more accurate simula-tion for the near-shore current jet observed alongthe Keweenaw coast in Lake Superior. Also, thisnewly-developed non-orthogonal model has suc-

FIG. 4. Numerical grids along the Keweenaw coast for the non-orthogonal coordinatetransformation (upper panel) and orthogonal coordinate transformation (lower panel).

50 Chen et al.

FIG. 5. Time series of observed (solid line) and model-predicted current speeds at thesurface of 8 or 10 m at buoys 1, 2, and 3 for July of 1973.

A Lake Superior Circulation Model 51

cessfully captured the spatial structure of the ther-mal current and Keweenaw Current measured dur-ing the 1999 KITES field measurement. This studyhas provided us with a good example of the valida-tion of this non-orthogonal model to the study ofthe near-shore current system.

ACKNOWLEDGMENTS

This research was supported by the National Sci-ence Foundation under grant number OCE-9712869and 0196543. We want to thank Joe Niebauer forsending us figures of observed currents taken nearthe Eagle Harbor in July of 1973 and for giving uspermission to use his data for the model-data com-parison. We also thank George Davidson at Univer-

sity of Georgia for his editorial help on this manu-script.

APPENDIX

The non-orthogonal coordinate transformationintroduced in our paper is different from the non-orthogonal coordinate model introduced in CH3Dor CH3D-WES. In our model, u1 and v1 are definedas the components of the velocity vector(see Fig. 1), where are the unit vectors tangent to curves ξ and η defined as

FIG. 6. Comparison between the satellite-derived and model-computed SST and along-shorecurrents. Left: the satellite-derived (upper) and model-predicted (lower) SST distribution alongthe Keweenaw coast on 17 September, 1999. Right: the ADCP measured (upper) and model-computed (lower) vertical profiles of the along-shore current at site E3 during 11–15 September1999.

! !e eξ η� and�

! !e eξ η� and�

!! !

!! !

ei j

ei jx y

x y

x yξ η

ξ ξ

ξ ξ

η η=

⋅ + ⋅

+=

⋅ + ⋅2 2

,� � � � �ηη ηx y

2 2+,� (A1)

52 Chen et al.

Substituting ξx, ξy, ηx, ηy into yields

where

By this definition, we can get a relationship be-tween (u1,v1) and (u, v) (the x and y components ofthe horizontal velocity on the orthogonal coordi-nate) as

This definition provides simple expressions of themomentum and tracer equations like those shown inan orthogonal coordinate model.

In CH3D or CH3D-WES, u1 and v1 are defined as

and

where� ( , ),� � ( , ),� � � ,� � �ξ ξ η η ξ ξη= = = = −x y x yy

Jx y

xx

Jy

J

x

JJ x y xx y

η

ξ ξξ η ηη η

,� � �

,� � � ,� � � �= − = = ⋅ −and ⋅⋅ yξ .

! !e eξ η� and�

! ! ! ! !e

hy i x j e

hy i xξ η η η ξ ξ= ⋅ − ⋅( ) = − ⋅ + ⋅1 1

2 1

,� � � � �! !j( ) (A2)

h x y h x y12 2

22 2= + = +ξ ξ η η,� .

uh

Jx u y v

vh

Jx u y v

12

11

= +

= +

( � � )

( � � ).

ξ ξ

η η

(A3)

! ! !

! ! !

iJ

y e y e

jJ

x e x e

= −

= − +

1

1

2 2

2 2

( )

( )

η η ξ η

η ξ ξ η

, (A8)

! ! !

! ! !e x i y j

e x i y jξ ξ ξ

η η η

2

2

= += +

( )

( );� � � � � � � � � .

!

!

e x y h

e x y h

ξ ξ ξ

η η η

22 2

1

22 2

2

= + =

= + =

(A9)

uJ

y u x v

vJ

y u x v

1

1

1

1

= −

= − +

( � � )

( � � )

η η

ξ ξ

(A5)

ud

dtv

d

dt1 1= =ξ η� and� . (A4)

J ui vj y e y e u x e x e( ) ( ) (! ! ! ! ! !+ = − + − +η ξ ξ η η ξ ξ η2 2 2 2 )) ,v (A7)

By this definition, (u1, v1) has a relationship with(u, v) as follows:

Let us define as the direction vectors of u1 and v1 in the non-orthogonal coordinates intro-duced in CH3D, and then

u = xξu1 + xηv1, v = yξu1 + yηv1. (A6)

According to the definition of a vector:

( ,� )! !e eξ η2 2

! ! ! ! !V u i v j u e v e i e= ⋅ + ⋅ = +1 2 1 2ξ ξ ,� . .,

so that

Therefore, are not the unit vectors! Because

are in the directions normal to ,respectively (see Fig. 7).

In the non-orthogonal coordinate defined inCH3D and CH3D-WES, all the force terms likepressure gradient forces, Coriolis forces, etc., aredivided into two terms in each component. Thismakes that model structure much more complicatedthan those in the orthogonal coordinate system and

! !e eξ η2 2� and�

! !e eξ η2 2� and�

! !e eη ξ� and�

! ! ! ! ! !

! !

e eh

y i x j x i y j

e e

ξ η η η η η

η

⋅ = −( ) ⋅ +( ) =

⋅

22

10

ξξ ξ ξ ξ ξ22

10= − +( ) ⋅ +( ) =

hy i x j x i y j

! ! ! !,

FIG. 7. Illustration of the non-orthogonal coor-dinate system used in CH3D or CH3D-WES andthe relationship between ( ) in CH3D and( ) in our model..

! !e � and� e2 2ξ η! !

e � and� eξ η

A Lake Superior Circulation Model 53

in the non-orthogonal coordinate introduced in thispaper. In addition, when the sigma-coordinate trans-formation is used in a stratified fluid, it leads tonumerical errors on sloping bottom topography dueto the inaccurate computation of the baroclinicpressure gradient force (Haney 1991, Chen andBeardsley 1995). Dividing the pressure gradientterms into two terms in each momentum equationprobably would make it more difficult to minimizethe sigma-coordinate error.

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Submitted: 8 May 2002Accepted: 7 May 2004Editorial handling: Martin T. Auer