JANUARY 2003 CHEN ET AL. 159

An Unstructured Grid, Finite-Volume, Three-Dimensional, Primitive Equations Model: Application to Coastal Ocean and Estuaries

CHANGSHENG CHEN AND HEDONG LIU School for Marine Science and Technology, University of Massachusetts±Dartmouth, New Bedford, Massachusetts

ROBERT C. BEARDSLEY Department of Physical , Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

(Manuscript received 18 January 2001, in ®nal form 27 June 2002)

ABSTRACT An unstructured grid, ®nite-volume, three-dimensional (3D) primitive equation ocean model has been devel- oped for the study of coastal oceanic and estuarine circulation. The model consists of momentum, continuity, temperature, salinity, and density equations and is closed physically and mathematically using the Mellor and Yamada level-2.5 turbulent closure submodel. The irregular bottom slope is represented using a ␴-coordinate transformation, and the horizontal grids comprise unstructured triangular cells. The ®nite-volume method (FVM) used in this model combines the advantages of a ®nite-element method (FEM) for geometric ¯exibility and a ®nite-difference method (FDM) for simple discrete computation. Currents, temperature, and salinity in the model are computed in the integral form of the equations, which provides a better representation of the conservative laws for mass, momentum, and heat in the coastal region with complex geometry. The model was applied to the Bohai , a semienclosed coastal ocean, and the Satilla River, a Georgia estuary characterized by numerous tidal creeks and inlets. Compared with the results obtained from the ®nite-difference model (ECOM-si), the new model produces a better simulation of tidal elevations and residual currents, especially around islands and tidal creeks. Given the same initial distribution of temperature in the Bohai Sea, the FVCOM and ECOM-si models show similar distributions of temperature and strati®ed tidal recti®ed ¯ow in the interior region away from the coast and islands, but FVCOM appears to provide a better simulation of temperature and currents around the islands, barriers, and inlets with complex topography.

1. Introduction ence model can provide a moderate ®tting of coastal Most of the world ' inner shelves and estuaries boundaries, but these transformations are incapable of are characterized by a series of barrier island complexes, resolving the highly irregular estuarine geometries char- inlets, and extensive intertidal salt marshes. Such an acteristic of numerous barrier island and tidal creek irregular geometric ocean±estuarine system presents a complexes (Blumberg 1994; Chen et al. 2001; Chen et challenge for oceanographers involved in model devel- al. 2002, manuscript submitted to J. Great Lakes Res.). opment even though the governing equations of oceanic The greatest advantage of the ®nite-element method is circulation are well de®ned and numerically solvable in its geometric ¯exibility. Triangular meshes at an arbi- terms of discrete mathematics. Two numerical methods trary size are used in this method and can provide an have been widely used in ocean models: 1) the ®nite- accurate ®tting of the irregular coastal boundary. The difference method (Blumberg and Mellor 1987; Haid- P-type ®nite-element method (Maday and Patera 1989) vogel et al. 1991; Blumberg 1994) and 2) the ®nite- or discontinuous Galerkin method (Reed and Hill 1973; element method (Lynch and Naimie 1993; Naimie Cockburn et al. 1990) has been introduced into the up- 1996). The ®nite-difference method is the simplest dis- dated ®nite-element model to help improve computa- crete scheme with an advantage of computational ef®- tional accuracy and ef®ciency. ciency. Introducing an orthogonal or nonorthogonal cur- Recently, the ®nite-volume method has received con- vilinear coordinate transformation into a ®nite-differ- siderable attention in the numerical computation of ¯uid dynamics (Dick 1994). The dynamics of oceanography comply with conservation laws. The governing equa- Corresponding author address: Dr. Changsheng Chen, School for tions of oceanic motion and water masses are expressed Marine Science and Technology, University of Massachusetts±Dart- mouth, 706 South Rodney French Blvd., New Bedford, MA 02744- by the conservation of momentum, mass, and energy in 1221. a unit volume. When the equations are solved numer- E-mail: [email protected] ically, these laws cannot always be guaranteed, espe-

᭧ 2003 American Meteorological Society

Unauthenticated | Downloaded 09/27/21 06:45 PM UTC 160 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 20 uץ uץ uץ uץ -cially in situations with sharp or discon ϩ u ϩ ␷ ϩ w Ϫ f ␷ zץ yץ xץ tץ -tinuous ¯ow. Unlike the differential form, the ®nite volume method discretizes the integral form of the equa- uץץ Pץ tions, making it easier to comply with the conservation 1 ϭϪ ϩ K ϩ F , (2.1) zץz muץ xץ laws. Since these integral equations can be solved nu- ␳ merically by the ¯ux calculation used in the ®nite-dif- o ΂΃ ␷ץ ␷ץ ␷ץ ␷ץ ference method over an arbitrarily sized triangular mesh ϩ u ϩ ␷ ϩ w ϩ fu zץ yץ xץ tץ like those in a ®nite-element method), the ®nite-volume) method seems to combine the best attributes of the ®- ␷ץץ Pץ nite-difference method (for simple discrete computa- 1 ϭϪ ϩ K ϩ F , (2.2) tional ef®ciency) and the ®nite-element method (for m ␷ zץz΂΃ץ yץ ␳o geometric ¯exibility). Pץ To our knowledge, a three-dimensional (3D), unstruc- ϭϪ␳g, (2.3) zץ tured grid, prognostic, primitive equation, ®nite-volume wץ ␷ץ uץ ocean circulation model is not currently available in the oceanographic community, although some efforts have ϩϩ ϭ0, (2.4) zץ yץ xץ been made to develop a ®nite-volume formulation of ␪ץ ␪ץ ␪ץ ␪ץ -the two-dimensional, barotropic shallow water equa ϩ u ϩ ␷ ϩ w zץ yץ xץ tץ tions (Ward 2000). The MIT General Circulation model developed by Marshall et al. (1997a,b) is the ®rst 3D ␪ץץ nite-volume ocean model. However, since this model® ϭ K ϩ F , (2.5) z ␪ץ z΂΃hץ -currently relies on rectangular structure grids for hori zontal discretization, it is not suited to use for coastal sץ sץ sץ sץ ocean and estuarine domains with complicated geom- ϩ u ϩ ␷ ϩ w zץ yץ xץ tץ etries. Recently, we have developed a 3D unstructured grid, ®nite-volume coastal ocean model (called sץץ FVCOM). This new model has been applied to the Bohai ϭ K ϩ F , (2.6) zץz΂΃hsץ ,Sea, a semienclosed coastal ocean, and the Satilla River a Georgia estuary characterized by numerous tidal ␳ ϭ ␳(␪, s), (2.7) creeks and inlets. Compared with results obtained from a well-developed ®nite-difference model (called where x, y, and z are the east, north, and vertical axes ECOM-si) and observational data, we ®nd that the ®- of the Cartesian coordinate; u, ␷, and w are the x, y, z nite-volume model provides a better simulation of tidal velocity components; ␪ is the potential temperature; s elevations and residual currents, especially around is- is the salinity; ␳ is the density; P is the pressure; f is lands and tidal creeks. Both FVCOM and ECOM-si the parameter; g is the gravitational accelera- show similar distributions of temperature and strati®ed tion; Km is the vertical viscosity coef®cient; and tidal recti®ed and buoyancy-induced ¯ows in the interior Kh is the thermal vertical eddy diffusion coef®cient. region in the Bohai Sea, but FVCOM seems to resolve Here Fu, F␷ , F␪, and Fs represent the horizontal mo- the detailed thermal structure and ¯ows around islands mentum, thermal, and salt diffusion terms. and complex coastal regions. Here Km and Kh are parameterized using the Mellor The remaining sections of this paper are organized and Yamada (1982) level-2.5 (MY-2.5) turbulent closure as follows. The model formulation, design of unstruc- scheme as modi®ed by Galperin et al. (1988). In the tured grids, and discretization procedure are described boundary layer approximation where the shear produc- in sections 2, 3, and 4, respectively. The model appli- tion of turbulent kinetic energy is produced by the ver- tical shear of the horizontal ¯ow near the boundary, the cations for the Bohai Sea and Satilla River are given 2 2 and discussed in section 5, and a summary is provided equations for q and q l can be simpli®ed as q 2ץq 2ץ qץq22ץ in section 6. Detailed expressions for the numerical ϩ u ϩ ␷ ϩ w zץ yץ xץ tץ -computation of individual terms in the momentum equa tion are given in an appendix. q2ץץ ϭ 2(Psbϩ P Ϫ␧) ϩ K qϩ F q, (2.8) zץz΂΃ץ The model formulation .2 ql 2ץql 2ץ qlץql22ץ a. The primitive equations ϩ u ϩ ␷ ϩ w zץ yץ xץ tץ The governing equations consist of the following mo- Ä 2 qlץץ mentum, continuity, temperature, salinity, and density W ϭ lE1 Psbϩ P Ϫ␧ϩ K qϩ F l, (2.9) zץ zץ equations: ΂΃΂΃E1

Unauthenticated | Downloaded 09/27/21 06:45 PM UTC JANUARY 2003 CHEN ET AL. 161 where q 2 ϭ (uЈ 2 ϩ ␷Ј 2)/2 is the turbulent kinetic energy; determined by matching a logarithmic bottom layer l is the turbulent macroscale; Kq is the vertical eddy to the model at a height z ab above the bottom; that is, diffusion coef®cient of the turbulent kinetic energy; Fq  and Fl represent the horizontal diffusion of the turbulent k2 22  kinetic energy and macroscale; Ps ϭ Km(uzzϩ ␷ ) and C ϭ max , 0.0025 , (2.14) d 2 Pb ϭ (gKh␳z)/␳o are the shear and buoyancy production zab terms of turbulent kinetic energy; ␧ϭq3/B l is the ln 1 ΂΃zo 2 turbulent kinetic energy dissipation rate; W ϭ 1 ϩ E 2l / ( L)2 is a wall proximity function, where LϪ1 ( ␬ ϭ ␨ Ϫ where k ϭ 0.4 is the von KaÂrmaÂn's constant and z is Ϫ1 (H z)Ϫ1; 0.4 is the von KaÂrmaÂn constant; o z) ϩ ϩ ␬ ϭ the bottom roughness parameter. H is the mean water depth; and is the free surface ␨ The surface and bottom boundary conditions for tem- elevation. In general, F and F are kept as small as q l perature are possible to reduce the effects of horizontal diffusion on ␪ 1ץ .the solutions The turbulent kinetic energy and macroscale equa- ϭ [Qn(x, y, t) Ϫ SW(x, y, ␨, t)], z ␳cKphץ tions are closed by de®ning at z (x, y, t), (2.15) Kmmϭ lqS , K hhϭ lqS , K qϭ 0.2lq. (2.10) ϭ ␨ ␪ץ The stability functions S and S are de®ned as m h ϭ 0, at z ϭϪH(x, y), (2.16) zץ 0.4275 Ϫ 3.354Gh Sm ϭ and (1 Ϫ 34.676Ghh)(1 Ϫ 6.127G ) where Q n (x, y, t) is the surface net heat ¯ux, which 0.494 consists of four components: downward shortwave Sh ϭ , (2.11) and longwave radiation, and sensible and latent ¯ux- 1 Ϫ 34.676Gh es; SW(x, y, 0, t) is the shortwave ¯ux incident at sea 2 2 surface; and c is the speci®c heat of . The where G h ϭ (l g/q ␳ o )␳ z . In the original MY level- p 2.5 turbulent closure model (Mellor and Yamada longwave radiation, and sensible and latent heat ¯uxes

1974, 1982), S m and S h are functions of the gradient are assumed here to occur at the ocean surface, while Richardson number. By removing a slight inconsis- the downward shortwave ¯ux SW(x, y, z, t) is ap- tency in the scaling analysis, Galperin et al. (1988) proximated by simpli®ed the MY turbulent closure model so that S m SW(x, y, z, t) and S h depend only on G h . Here G h has an upper bound of 0.023 for the case of unstable (␳ z Ͼ 0) strat- ϭ SW(x, y, 0, t)[Rez/azϩ (1 Ϫ R)e /b], (2.17) i®cation and a lower bound of Ϫ0.28 for the case of stable (␳ z Ͻ 0) strati®cation. Parameters A1 ,A2 ,B1 , where a and b are attenuation lengths for longer and B 2 , and C1 are given as 0.92, 16.6, 0.74, 10.1, and shorter (blue-green) wavelength components of the 0.08, respectively. shortwave irradiance, and R is the percent of the total The surface and bottom boundary conditions for u, ¯ux associated with the longer wavelength irradiance. ␷, and w are This absorption pro®le, ®rst suggested by Kraus (1972), has been used in numerical studies of upper- ␷ 1 ocean diurnal heating by Simpson and Dickeyץ uץ Kmsxsy, ϭ (␶ , ␶ ), z ␳ o (1981a,b) and others. The absorption of downwardץ zץ΂΃ irradiance is included in the temperature (heat) equa- ␨ץ ␨ץ ␨ץ w ϭϩu ϩ ␷ , tion in the form of yץ xץ tץ (SW(x, y, z, tץ at z ϭ ␨(x, y, t), and (2.12) HÃ (x, y, z, t) ϭ zץ ␷ 1ץ uץ K , ϭ (␶ , ␶ ), mbxbyz z SW(x, y, 0, t) R 1 Ϫ R (␳ o ϭ ez/a ϩ ez/b . (2.18 ץ ץ΂΃ ␳ca b H p ΂΃ץ Hץ w ϭϪu Ϫ ␷ , y This approach leads to a more accurate prediction ofץ xץ near-surface temperature than the ¯ux formulation based at z ϭϪH(x, y), (2.13) on a single wavelength approximation (Chen et al. 22 where (␶ sx, ␶ sy ) and (␶ bx , ␶ by ) ϭ C d͙u ϩ ␷ (u, ␷) 2002). are the x and y components of surface and bot- The surface and bottom boundary conditions for sa- tom stresses; D ϭ H ϩ ␨. The drag coef®cient C d is linity are

Unauthenticated | Downloaded 09/27/21 06:45 PM UTC 162 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 20 sץ ␪ץ (ss(PÃÃϪ Eץ ϭ at z ϭ ␨(x, y, t) and ␷ n ϭ 0; ϭ 0; ϭ 0, (2.22) nץ nץ zKh ␳ץ -s where ␷ is the velocity component normal to the boundץ ϭ 0atz ϭϪH(x, y), (2.19) n .z ary, and n is the coordinate normal to the boundaryץ where PÃ and EÃ are precipitation and evaporation rates, respectively. Note that a groundwater ¯ux can be easily b. The governing equations in the ␴ coordinate added into the model by modifying the bottom boundary conditions for vertical velocity and salinity. The ␴-coordinate transformation is used in the ver- The surface and bottom boundary conditions for the tical in order to obtain a smooth representation of ir- turbulent kinetic energy and macroscale equations are regular bottom topography. The ␴-coordinate transfor- 222/32 qlϭ 0, q ϭ B1 u␶s at z ϭ ␨(x, y, t), (2.20) mation is de®ned as

222/32 qlϭ 0, q ϭ B1 u␶b at z ϭϪH(x, y), (2.21) z Ϫ ␨ z Ϫ ␨ ␴ ϭϭ, (2.23) where u␶s and u␶b are the friction velocities associated H ϩ ␨ D with the surface and bottom stresses. The kinematic and heat and salt conditions on the where ␴ varies from Ϫ1 at the bottom to 0 at the surface. solid boundary are speci®ed as In this coordinate, equations (2.1)±(2.9) are given as

D 0ץץ ␨ gDץ u␻ץ u␷Dץ uD2ץ uDץ ϩϩϩϪf ␷D ϭϪgD Ϫ D ␳ d␴Јϩ␴␳ xץ ͵ xץ x ␳oץ ␴ץ yץ xץ tץ []΂΃␴ uץץ 1 ϩ K ϩ DF , (2.24) ␴ץ␴΂΃mxץ D D 0ץץ ␨ gDץ ␷␻ץ ␷ 2Dץ u␷Dץ ␷Dץ ϩϩϩϩfuD ϭϪgD Ϫ D ␳ d␴Јϩ␴␳ yץ ͵ yץ y ␳oץ ␴ץ yץ xץ tץ []΂΃␴ ␷ץץ 1 ϩ K ϩ DF , (2.25) ␴ץ␴΂΃myץ D ␪ץץ ␪␻ 1ץ ␪␷Dץ ␪uDץ ␪Dץ ϩϩϩϭ K ϩ DHÃ ϩ DF , (2.26) ␴ ␪ץ ␴΂΃hץ ␴ Dץ yץ xץ tץ sץץ s␻ 1ץ s␷Dץ suDץ sDץ ϩϩϩϭ K ϩ DF , (2.27) ␴ץ␴΂΃hsץ ␴ Dץ yץ xץ tץ q 2ץץ q 2␻ 1ץ q 2␷Dץ quDץqD22ץ ϩϩϩϭ2D(P ϩ P Ϫ␧) ϩ K ϩ DF , (2.28) ␴ qץ␴΂΃ qץ ␴ sbDץ yץ xץ tץ ql 2ץץ ql 2␻ WÄ 1ץ ql 2␷D ␻ץ qluDץqlD22ץ ϩϩϩ ϭlE1 D Psbϩ P Ϫ␧ϩ K q ϩ DF1, (2.29) ␴ץ ␴ץ ␴ ΂΃΂΃ED1ץyDץ xץ tץ ␳ ϭ ␳(␪, s). (2.30)

The horizontal diffusion terms are de®ned as D(F␪, Fsq, F 22, F ql) ␷ץ uץץ uץץ ץץץץ (DF 2AH ϩ AH ϩ , (2.32 xmഠ m ഠ AH ϩ AH (␪, s, q22, ql), (2.34) yץ yץ xץx΂΃΂΃hhץ[] [ xץ yץy ΂΃ץ xץ][]xץ ␷ where A and A are the horizontal eddy and thermalץץ ␷ץ uץץ DF ഠ AH ϩϩ2AH , (2.33) m h -y diffusion coef®cients, respectively. Following the ®niteץy mץ xץ yץx[][]΂΃ץym

Unauthenticated | Downloaded 09/27/21 06:45 PM UTC JANUARY 2003 CHEN ET AL. 163 difference primitive equation coastal ocean models c. The 2D (vertically integrated) equations (called POM and ECOM-si) developed originally by Blumberg and Mellor (1987), this de®nition ensures the The sea surface elevation included in the equations validity of the bottom boundary layer simulation in the describes the fast-moving surface gravity waves. In the ␴-coordinate transformation system. The detailed de- explicit numerical approach, the criterion for the time scription was given in Mellor and Blumberg (1985). step is inversely proportional to the phase speed of these The boundary conditions are given as follows. At the waves. Since the sea surface elevation is proportional surface where ␴ ϭ 0, to the gradient of water transport, it can be computed using vertically integrated equations. The 3D equations ␷ D then can be solved under conditions with a given seaץ uץ , ( , ), 0, ϭ ␶ sx␶ sy ␻ ϭ surface elevation. In this numerical method, called ␴␳omKץ ␴ץ΂΃ ``mode splitting,'' the currents are divided into external ␪ D and internal modes that can be computed using twoץ ϭ [Qn(x, y, t) Ϫ SW(x, y, 0, t)], ␴␳cKph distinct time steps. This approach is used successfullyץ in POM. ss(P Ϫ E)Dץ ϭϪ , ql2 ϭ 0, Recently, a semi-implicit scheme was introduced into ␴ Kh POM, in which the sea surface elevation was computedץ implicitly using a preconditioned conjugate gradient q22/32Bu, (2.35) ϭ 1 ␶s method with no sacri®ce in computational time (Casulli and at the bottom where ␴ ϭϪ1 and Cheng 1991). This updated version of POM is called ECOM-si. The semi-implicit scheme cannot easily be ␷ D applied to a ®nite-volume model since it is dif®cult toץ uץ , ϭ (␶ bx, ␶ by), ␻ ϭ 0, construct a linear positive symmetric algebraic matrix ␴␳Kץ ␴ץ ΂΃om when unstructured triangular meshes are used. For this s reason, we select the mode-splitting method to solve theץ ␪ץ ϭϭ0, ql2 ϭ 0, .␴ momentum equationsץ ␴ץ The 2D (vertically integrated) momentum and con- 22/32 q ϭ B1 u␶b. (2.36) tinuity equations are given as

(␷ D)ץ (uD)ץ ␨ץ ϩϩϭ0, (2.37) yץ xץ tץ

D 0ץץ␨ gD 00ץ u␷ Dץ uD2ץ uDץ ϩϩ Ϫf ␷ D ϭϪgD Ϫ D ␳ d␴Ј d␴ ϩ ␴␳ d␴ ͵ xץ xץ͵͵ x ␳oץ yץ xץ tץ Ά Ϫ1 ΂΃␴ Ϫ1 ·

Ϫsx␶ bx Äץ ϩϩDFxxϩ G , (2.38) ␳o

D 0ץץ␨ gD 00ץ ␷ 2Dץ u␷ Dץ ␷ Dץ ϩϩϩfuDϭϪgD Ϫ D ␳ d␴Ј d␴ ϩ ␴␳ d␴ ͵ yץ yץ͵͵ y ␳oץ yץ xץ tץ Ά Ϫ1 ΂΃␴ Ϫ1 ·

␶ syϪ ␶ by Ä ϩϩDFyyϩ G , (2.39) ␳o

␷ 2Dץ u␷ Dץ where G and G are de®ned as x y G ϭϩϪDFÄ yץ xץyy u␷ Dץ uD2ץ ␷ 2Dץ u␷ Dץ G ϭϩ ϪDFÄ (y ϪϩϪDF , (2.41ץ xץxx y yץ xץ[] u␷ Dץ uD2ץ ϪϩϪDF , (2.40) and the horizontal diffusion terms are approximately y x given asץ xץ[]

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FIG. 1. The unstructured grid for the ®nite-volume model.

(␷ [Xnn( j), Y ( j)], j ϭ 1:M. (3.2ץ uץץ uץץ DFÄ ഠ 2AH ϩ AH ϩ , (2.42) x ] Since none of the triangles in the grid overlap, N shouldץ yץy m΂΃ץ xץ][]xץxm also be the total number of unstructured triangles. On each triangle cell, the three nodes are identi®ed using ␷ץץ ␷ץ uץץ Ä DFymഠ AH ϩϩ2AH m, (2.43) integral numbers de®ned as N ( ˆj), where ˆj is counted y iץ yץ xץ yץx[][]΂΃ץ clockwise from 1 to 3. The surrounding triangles that have a common side are counted using integral numbers ␷ de®ned as NBE ( ˆj), where ˆj is counted clockwise fromץ uץץ uץץ DFxmഠ 2AH ϩ AH mϩ , (2.44) i x 1 to 3. At open or coastal solid boundaries, NBE ( ˆj)isץ yץy ΂΃ץ xץ xץ i speci®ed as zero. At each node, the total number of the ␷ surrounding triangles with a connection to this node isץץ ␷ץ uץץ DF ഠ AH ϩϩ2AH . (2.45) y expressed as NT( j), and they are counted using integralץy mץ xץ yץx ΂΃ץym numbers NBi(m), where m is counted clockwise from 1 The overbar `` '' denotes the vertically integration. For to NT( j). example, for a given variable ␺, To provide a more accurate estimation of the sea sur-

0 face elevation, currents, and salt and temperature ¯uxes, ␺ ϭ ␺ d␴. (2.46) the numerical computation is conducted in a specially ͵ designed triangular grid in which , ,s, , ,q2, q 2l, Ϫ1 ␨ ␻ ␪ ␳

H, D, Km, Kh, Am, and Ah are placed at nodes, and u, ␷ are placed at centroids. Variables at each node are de- 3. Design of the unstructured grids termined by a net ¯ux through the sections linked to centroids in the surrounding triangles with connection Similar to the ®nite-element method, the horizontal to that node. Variables at centroids are calculated based numerical computational domain is subdivided into a on a net ¯ux through three sides of that triangle. The set of nonoverlapping unstructured triangular cells. An numerical code was written using Fortran 77 and can unstructured triangle comprises three nodes, a centroid, be run on a PC or workstation with Fortran 77 or above. and three sides (Fig. 1). Let N and M be the total number of centroids and nodes in the computational domain, respectively, then the locations of centroids can be ex- 4. The discretization procedure pressed as a. The 2D external mode [X(i), Y(i)], i ϭ 1:N; (3.1) Let us consider the continuity equation ®rst. Inte- and the locations of nodes can be speci®ed as grating Eq. (2.37) over a given triangle area yields

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FIG. 2. Geometry of the Bohai Sea. Filled dots with numbers 1±32 shown along the coast are tidal measurement stations. Two heavy solid lines and ®lled triangles are the sections and site used for model comparisons.

0 n , ␷ D) ␨jjϭ ␨)ץ (uD)ץ ␨ץ dx dy ϭϪ ϩ dx dy (x NT( jץ xץ ͵͵ tץ͵͵ [] 0 nnnn R ϭ R ϭ [(⌬x ␷ Ϫ⌬yu)D ␨␨͸ 2mϪ1 m 2mϪ1 m 2mϪ1 mϭ1

ϭϪ ␷ nDdsЈ, (4.1) nnn ϩ (⌬x2mm␷ Ϫ⌬yu2mm)D2m], (4.2) Ͷs Ј ⌬tRkϪ1 where ␷ is the velocity component normal to the sides ␨k ϭ ␨0 Ϫ ␣k ␨ , and ␨nϩ14ϭ ␨ , (4.3) n jj␨ j j of the triangle and sЈ is the closed trajectory that com- 2⍀j prises the three sides. Equation (4.1) is integrated nu- where k ϭ 1, 2, 3, 4 and (␣1, ␣ 2, ␣3, ␣ 4) ϭ (1/4, 1/3, merically using the modi®ed fourth-order Runge±Kutta 1/2, 1). Superscript n represents the nth time step. Here ␨ time-stepping scheme. This is a modi®ed multistage ⍀j is the area enclosed by the lines through centroids time-stepping approach with second-order accuracy and midpoints of side of surrounding triangles con- n (Dick 1994). The detailed procedure for this method is nected to the node where ␨j is located. Alsou m , and n described as follows: ␷ m are de®ned as

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FIG. 3. Geometry of the Satilla River. Water depth contours are in meters. Filled dots are seven bottom pressure measurement sites conducted by Dr. Blanton at Skidaway Institute of Ocean- ography, Savannah, GA.

nn nn u mmϭ u[NT(m)] , ␷ ϭ ␷[NT(m)] . (4.4) ⌬y2mϪ12ϭ y m Ϫ y2mϪ12, ⌬y m ϭ y2mϩ12Ϫ y m. (4.6) The time step for the external mode is ⌬t and Similarly, integrating Eqs. (2.38) and (2.39) over a given ⌬x2mϪ12ϭ x m Ϫ x2mϪ12, ⌬x m ϭ x2mϩ12Ϫ x m, (4.5) triangle area, we get

␨ץ uDץ dx dy ϭϪ uD␷ dsЈϩ f ␷ DdxdyϪ gD dx dy xץ ͵͵ ͵͵ t nץ ͵͵ Ͷs Ј

Dץ0␳␴ ץץgD2 00 Ϫ ␳ d␴ Ϫ d␴ d␴ dx dy xץ ␴ Dץ ͵ ͵ xץ͵ ␳o͵͵ Ά·Ϫ1 []␴␴

␶ sxϪ ␶ bx Ä ϩ dx dy ϩ DFx dx dy ϩ Gdxdy,x (4.7) ͵͵␳o ͵͵ ͵͵

␨ץ ␷ Dץ dx dy ϭϪ ␷ D␷ n dsЈϪ fuDdxdyϪ gD dx dy yץ ͵͵ ͵͵ t ͶsЈץ͵͵

Dץ0␳␴ ץץgD2 00 Ϫ ␳ d␴ Ϫ d␴ d␴ dx dy yץ ␴ Dץ ͵ ͵ yץ͵ ␳o͵͵ Ά·Ϫ1 []␴␴

␶ syϪ ␶ by Ä ϩ dx dy ϩ DFy dx dy ϩ Gdxdy.y (4.8) ͵͵␳o ͵͵ ͵͵

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FIG. 4. Unstructured and curvilinear grids of the Bohai Sea for FVCOM and ECOM-si.

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FIG. 5. The charts of model-predicted coamplitudes and cophases of the M2 and S2 in the Bohai Sea. (right) ECOM-si and (left) FVCOM.

Equations (4.7) and (4.8) are also integrated numerically where the de®nitions of k and ␣k are the same as those u ␷ using the modi®ed fourth-order Runge±Kutta time-step- shown in Eq. (4.3). Here⍀⍀ii and are the triangle ping scheme as follows: areas whereu and␷ are located. In the grids used in 0 n 0 n 0 n this model,u and␷ are all at the centroid, so that ⍀u u iiϭ u , ␷ iiϭ ␷ , R uuϭ R , i 0 n ϭϭ⍀⍀␷ . The depth D is at the centroid, which is R ϭ R , (4.9) i i i ␷␷ interpolated from depth values at three nodes. Here 0 nn ⌬tRu RRand represent all the terms on the right of Eqs. u k ϭ u 0 Ϫ ␣k , u ␷ ii u 4⍀iiD (4.7) and (4.8), respectively. They are equal to 0 ⌬tR k 0 k ␷ n ␷ iiϭ ␷ Ϫ ␣ , (4.10) ␷ R u ϭ ADVU ϩ DPBPX ϩ DPBCX ϩ CORX 4⍀iiD nϩ14 nϩ14 u iiiiϭ u , ␷ ϭ ␷ , (4.11) ϩ VISCX Ϫ Gx, (4.12)

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FIG. 6. The charts of model-predicted coamplitudes and cophases of the K1 and O1 tides in the Bohai Sea. (right) ECOM-si and (left) FVCOM.

n R␷ ϭ ADVV ϩ DPBPY ϩ DPBCY ϩ CORY b. The 3D internal mode

ϩ VISCY Ϫ Gy, (4.13) The momentum equations are solved numerically us- ing a simple combined explicit and implicit scheme in which the local change of the currents is integrated using where ADVU and ADVV, DPBPX and DPBPY, DPBCX the ®rst-order accuracy upwind scheme. The advection and DPBCY, CORX and CORY, VISCX and VISCY are terms are computed explicitly by a second-order ac- the x and y components of vertically integrated hori- curacy Runge±Kutta time-stepping scheme is also in- zontal advection, barotropic pressure gradient force, corporated in the updated version to increase the nu- , and horizontal diffusion terms, respec- merical integration to second-order accuracy. The pro- tively. The de®nitions of Gx and Gy are the same as cedure for this method is very similar to that described those shown in Eqs. (2.40) and (2.41). The numerical above for the 2D external mode. To provide a simple approach for these terms is given in the appendix. interpretation of the numerical approach for the 3D in-

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FIG. 8. Distributions of model-predicted near-surface (1/2 ␴-level

below the surface) temperature averaged over a M2 tidal cycle of the FIG. 7. Distributions of (a) FVCOM and (b) ECOM-si predicted 10th model day. (top) FVCOM and (bottom) ECOM-si. surface residual current vectors around the islands close to the Bohai Sea Strait.

R␷ ϭ ADVV3 ϩ CORY3 ϩ DPBPY3 ϩ BPBCY3 ϩ HVISCY. (4.16) ternal mode, we focus our description here only on the The numerical integration is conducted in two steps. In ®rst-order accuracy upwind scheme. It should be noted the ®rst step, the ``transition'' velocity is calculated using here that the second-order accuracy Runge±Kutta time- all the terms except the vertical diffusion term in the mo- stepping scheme is also incorporated in the model. mentum equations. Then the true velocity is determined The 3D momentum equations can be rewritten as implicitly using a balance between the local change of the ``transition'' velocity and the vertical diffusion term. uץץ uD 1ץ ϩ Rumϭ K , Letu*i,k and␷* i,k be the x and y components of the ␴ ``transition'' velocity at the midpoint between the k andץ␴΂΃ץtDץ k ϩ 1 ␴-levels in triangular cell i. They can be deter- ␷ץץ ␷D 1ץ ϩ R ϭ K , (4.14) mined numerically as follows: ␴ץ ␴ mץ tD␷ץ ΂΃ ⌬t n I n where u*i,kϭ u i,kϪ R u,(i,k), ⍀⌬ii␴D

Ru ϭ ADVU3 ϩ CORX3 ϩ DPBPX3 ⌬t n I n ␷*i,kϭ ␷ i,k Ϫ R␷,(i,k), (4.17) ϩ BPBCX3 ϩ HVISCX, (4.15) ⍀⌬ii␴D

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FIG. 9. Vertical distributions of temperature averaged over a M 2 tidal cycle of the 10th model day on cross-sea sections 1 and 2. (right) ECOM-si and (left) FVCOM. where ⌬␴ ϭ ␴ Ϫ ␴ , and ⌬t is the time step for the 2K (k)⌬t k kϩ1 I C ϭϪ m , internal mode. The numerical approach for computing i,k nϩ12 [D ](␴kkϪ ␴ ϩ1)(␴kϪ1 Ϫ ␴kϩ1) nn RRu,(i,k) and␷,(i,k) is described in detail in the appendix. After the transition velocity is determined, the true Bi,kϭ 1 Ϫ A i,kϪ C i,k. (4.20) nϩ1 nϩ1 velocity (ui,k and␷ i,k ) at the (n ϩ 1)th time step can Equations (4.18) and (4.19) are tridiagonal equations, be found by solving the following discrete equation: which can be solved easily for given surface and bottom boundary conditions. nϩ1 nϩ1 nϩ1 Aui,k i,kϩ1 ϩ Bui,k i,kϩ Cu i,k i,kϪ1 ϭ u*i,k , (4.18) A similar numerical approach is also used to solve the equations for ␪,s,q2, and q 2l. For example, the A ␷ nϩ1 ϩ B ␷ nϩ1 ϩ C ␷ nϩ1 ϭ ␷* , (4.19) i,k i,kϩ1 i,k i,k i,k i,kϪ1 i,k temperature equation can be rewritten as where ␪ץץ ␪D 1ץ ϩ R ϭ K , (4.21) ␴ץ ␴ hץ tD␪ץ 2K (k ϩ 1)⌬t A ϭϪ m , ΂΃ i,k nϩ12 [D ](␴kkϪ ␴ ϩ1)(␴kkϪ ␴ ϩ2) where

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␪␻Dץ ␪␷Dץ ␪uDץ R ϭϩϩ ϪDHÃ Ϫ DF . (4.22) ␴ץ yץ xץ␪␪ Equation (4.21) is identical to the equations for u and

␷ in Eq. (4.14) if u or ␷ is replaced by ␪,Ru or R␷ by R␪, and Km by K␪. The only difference is that ␪ is cal- culated at nodes and has the same control volume as that used for ␨. To apply a second-order upwind scheme for the temperature advection term, we used Green's theorem to calculate the temperature gradient at nodes (Barth 1993; Wu and Bogy 2000). Computing ␪ at nodes has shown a signi®cant improvement in the advective temperature ¯ux over steep bottom topography. A de- tailed discussion about this issue will be given in a separate manuscript. The detailed description of the nu- merical approach for Eq. (4.21) is given in the appendix.

5. Model applications To test our new unstructured grid, ®nite-volume, ocean circulation model, we applied it to the Bohai Sea around the northern coast of China and the Satilla River in the inner shelf of the South Atlantic Bight. The Bohai Sea is a semienclosed coastal ocean that includes mul- tiple islands and coastal inlets (Fig. 2). The mean depth of the Bohai is about 20 m, with the deepest region of about 70 m located near the northern coast of the Bohai Strait. The Satilla River is a typical estuary character- ized by complex curved coastlines, multiple tidal creeks and inlets (Fig. 3). The mean depth of this river is about 4 m, with the deepest region being about 20 m near the river mouth. In the Bohai Sea, the motion is dominated by semi- diurnal (M2 and S2) and diurnal (K1 and O1) tides, which FIG. 10. Time series of model-predicted temperature at the surface account for about 60% of the current variation and ki- (1/2 ␴ level below the surface), middle-depth, and bottom (1/2 ␴ netic energy there. Since the tidally recti®ed residual level above the bottom) at selected sites I and II (shown in Fig. 2) ¯ow is only substantial near the coast and islands in the in the Bohai Sea. Solid line: FVCOM; dashed line: ECOM-si. Bohai Sea, geometric ®tting is essential to providing a more accurate simulation of the tidal waves and residual variations of biological and chemical materials in this ¯ow. The Bohai Sea is connected to the Yellow Sea (on estuary (Bigham 1973; Dunstan and Atkinson 1976; the south) through the Bohai Strait. Several islands lo- Pomeroy et al. 1993; Verity et al. 1993; Zheng and Chen cated in the Strait complicate the water exchange be- 2000). Since the Satilla River estuary features numerous tween these two . Failing to resolve these islands tidal creeks, failing to resolve these creeks would lead leads to an underestimation of water transport through to under- or overestimating the tidally recti®ed ¯ow. the strait. It also results in an unrealistic distribution of This in turn would cause water transport in the river to the tidal motion in the Bohai Sea due to alterations in be miscalculated. This can be seen clearly in the com- the propagation paths of tidal waves. In addition, in the parison between the ®nite-difference and ®nite-volume Bohai Sea, the tidally recti®ed residual ¯ow is usually model results of the Satilla River given below. one order of magnitude smaller than the buoyancy- and wind-induced ¯ows, except near the coast and around islands. In order to obtain a more accurate simulation a. The Bohai Sea of temperature and salinity, the model must be able to The ®nite-difference model used in this comparison resolve the complex topography near the coast and is ECOM-si, which is an updated version of POM. The around islands. model domains for FVCOM and ECOM-si are shown

In the Satilla River, the M2 tidal current accounts for in Fig. 4, both of which have their open boundaries in about 90% of the along-river current variation (Blanton the Yellow Sea about 150 km south of the Bohai Strait. 1996). Tidal advection and mixing also are the main In FVCOM, the horizontal resolution is about 2.6 km physical processes controlling the spatial and temporal around the coast and about 15±20 km in the interior and

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FIG. 12. Comparison between model-predicted (FVCOM and

ECOM-si) and observed amplitudes of M2 tidal elevation at seven measurement sites shown in Fig. 3.

value of 30 psu. The time step was 186.3 s, which cor-

responded to 240 time steps over the M2 tidal cycle. The model-predicted time series of surface elevation and currents at each grid point was ®tted by a least squares harmonic analysis method. The resulting coam- plitude and cophase of each tidal constituent are shown

FIG. 11. Unstructured and curvilinear grids of the Satilla River for FVCOM and ECOM-si.

near the open boundary. In ECOM-si, a uniform hori- zontal resolution of about 2 km is used in most of the computational areas except near the open boundary where the horizontal resolution is about 7 km. In the vertical, both FVCOM and ECOM-si models comprise ten uniformly distributed ␴ layers, which result in a vertical resolution of about 0.1±1.0 m in the coastal region shallower than 10 m, and about 6 m at the 60- m isobath. The models were driven using the same semi- diurnal (M2 and S2) and diurnal (O1 and K1) tidal el- evations and phases at the open boundary. The data used for tidal forcing were interpolated directly from our East China/Yellow Seas model and adjusted according to previous tidal measurements at the northern and southern coasts. To examine each model's capability of simulating buoyancy-induced currents, we ran both models prognostically using the same initial strati®ca- tion. The initial temperature was speci®ed as a vertical linear function with 25ЊC at the surface and 15ЊCata FIG. 13. Distributions of the near-surface M2 tidal currents at the depth of 75 m. The salinity was speci®ed as a constant maximum ¯ood . (top) FVCOM and (bottom) ECOM-si.

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FIG. 14. Distributions of the near-surface M2 tidal currents at the maximum ebb tide. (top) FVCOM and (bottom) ECOM-si. in Figs. 5 and 6. The model±data comparisons of tidal amplitudes and phases at tidal measurement stations are FIG. 15. Distribution of model-predicted surface residual current given in Tables 1±4. Although both FVCOM and vectors in the selected area of the Satilla River. (top) FVCOM and (bottom) ECOM-si. ECOM-si show that M2 and S2 tidal waves propagate counterclockwise around the coast like a , the distributions of tidal amplitudes and phases pre- dicted by these two models differ signi®cantly. FVCOM stituent in Bohai Bay and Liaodong Bay. The standard predicts two nodes of the M2 and S2 tides in the Bohai deviation for the M2 and S2 tidal simulations is 6.0 and Sea: one is near the mouth of the Yellow River on the 5.8 cm in amplitude and 18.9Њ and 29.9Њ in phase, re- southwestern coast, and the other is located offshore of spectively, in the case with FVCOM. However, they are Qinhuangdao on the northwestern coast. These two 16.6 and 5.9 cm in amplitude and 41.2Њ and 42.9Њ in nodes, however, shift onshore in the case of the ECOM- phase in the case with ECOM-si (Tables 1 and 2). The si, especially for the M2 tide. The FVCOM-predicted FVCOM is more capable of predicting the amplitude maximum amplitudes of the M2 and S2 tides are about and phase of semidiurnal tides in Liaodong Bay and 130 cm in Liaodong Bay and 100 cm in Bohai Bay, Bohai Bay than ECOM-si. However, no signi®cant dif- both of which are about 10±20 cm higher than those ferences are found for the K1 and O1 tides in both the predicted by the ECOM-si. Both FVCOM and ECOM- FVCOM and ECOM-si models (Tables 3 and 4). si show similar structures for the K1 and O1 tides, but Both FVCOM and ECOM-si predict relatively weak the model-predicted amplitude of the K1 tide is higher tidally recti®ed residual currents in the Bohai Sea except in the case with FVCOM than in the case with ECOM- near the coast and around islands. In the Bohai Strait, si. for example, the FVCOM model shows multiple around- The comparison between observed and model-pre- island residual ¯ow patterns. These patterns are not well dicted amplitudes and phases of semidiurnal tides at predicted by the ECOM-si model because of poor res- tidal measurement stations around the Bohai Sea shows olution around the islands (Fig. 7). Although both a better agreement in the case with FVCOM than in the FVCOM and ECOM-si models show an eastward re- case with ECOM-si, especially for the M2 tidal con- sidual ¯ow along the southern coast in the Bohai Sea,

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TABLE 1. Model±data comparison of the M2 tidal amplitude and phase in the Bohai Sea. Amplitude (cm) Phase (ЊG) FVCOM ECOM-si FVCOM ECOM-si

Stations ␨o ␨c ⌬␨ ␨c ⌬␨ ␾o ␾c ⌬␾ ␾c ⌬␦ 1 88.0 89.2 Ϫ1.2 83.3 4.7 294.0 293.7 0.3 294.3 Ϫ0.3 2 83.0 80.5 2.5 79.9 3.1 300.0 301.8 Ϫ1.8 300.8 Ϫ0.8 3 59.0 53.0 6.0 54.5 4.5 350.0 351.6 Ϫ1.6 350.3 Ϫ0.3 4 61.0 57.6 3.4 60.4 0.6 11.0 9.6 1.4 14.8 Ϫ3.8 5 54.0 55.1 Ϫ1.1 56.5 Ϫ.2.5 82.0 82.7 Ϫ0.7 77.5 4.5 6 71.0 73.3 Ϫ2.3 68.1 2.9 112.0 101.7 10.3 96.1 15.9 7 120.0 122.8 Ϫ2.8 112.3 7.8 125.0 125.9 Ϫ0.9 120.8 4.2 8 116.0 124.9 Ϫ8.9 107.3 8.7 145.0 145.9 Ϫ0.9 143.5 1.5 9 96.0 106.3 Ϫ10.3 82.3 13.7 150.0 155.7 Ϫ5.7 149.1 0.9 10 42.0 40.1 1.9 49.2 Ϫ7.2 153.0 164.3 Ϫ1.3 153.5 Ϫ0.5 11 25.0 28.0 Ϫ3.0 13.6 11.4 162.0 172.3 Ϫ0.3 155.2 6.8 12 13.0 20.9 Ϫ7.9 6.1 6.9 170.0 185.2 Ϫ5.2 159.2 10.8 13 12.0 15.6 Ϫ3.6 6.1 5.9 179.0 192.6 Ϫ3.6 159.2 19.8 14 5.0 11.1 Ϫ6.1 11.4 Ϫ6.4 282.0 269.4 12.6 332.0 Ϫ50.0 15 11.0 15.6 Ϫ4.6 17.8 Ϫ6.8 311.0 291.8 19.2 337.2 Ϫ26.2 16 17.0 22.5 Ϫ5.5 24.5 Ϫ7.5 339.0 317.5 21.5 343.1 Ϫ4.1 17 32.0 31.5 0.5 24.5 7.5 0.0 341.4 18.6 343.1 16.9 18 73.0 66.2 6.8 54.1 18.9 74.0 67.2 6.8 66.8 7.2 19 12.10 106.1 14.9 77.1 43.9 85.0 80.7 4.3 79.8 5.2 20 117.0 111.2 5.8 77.1 39.9 90.0 86.3 3.7 79.8 10.2 21 106.0 117.1 Ϫ11.1 94.9 11.1 96.0 99.6 Ϫ3.6 116.4 Ϫ20.4 22 112.0 106.6 5.4 94.9 17.1 109.0 110.2 Ϫ1.2 116.4 Ϫ7.4 23 84.0 75.5 8.5 56.1 27.9 127.0 114.1 12.9 130.4 Ϫ3.4 24 49.0 50.9 Ϫ1.9 30.0 19.0 313.0 322.1 Ϫ9.1 124.3 Ϫ71.3 25 46.0 50.3 Ϫ4.3 58.5 Ϫ12.5 230.0 322.2 Ϫ2.2 353.7 Ϫ23.7 26 48.0 50.4 Ϫ2.4 54.0 Ϫ6.0 320.0 330.2 Ϫ0.2 3.4 Ϫ43.4 27 51.0 45.5 5.5 40.8 10.2 319.0 335.0 Ϫ6.0 3.2 Ϫ44.2 28 40.0 32.4 7.6 22.5 17.5 316.0 324.0 Ϫ8.0 340.4 Ϫ24.4 29 60.0 55.1 4.9 49.3 10.7 300.0 300.5 Ϫ0.5 297.4 2.6 30 54.0 47.9 6.1 43.1 10.9 296.0 296.6 Ϫ0.6 294.4 1.6 31 74.0 69.7 4.3 32.2 41.8 290.0 283.4 6.6 282.1 7.9 32 76.0 74.1 1.9 73.9 2.1 290.0 284.6 5.4 277.4 12.6 Std dev 6.0 16.6 18.9 41.2

Note: ␨oÐobserved amplitude; ␨cÐcomputed amplitude; ⌬␨ ϭ ␨o Ϫ ␨c; ␾o±observed phase; ␾cÐcomputed phase; and ⌬␾ ϭ ␾o Ϫ ␾c. the current is trapped near the coast and is much stronger two models. At site I (in the interior), for example, both in the case with FVCOM than in the case with ECOM- FVCOM and ECOM-si show that the temperature at the si. Similar disparities also are found around the islands surface and middle depth remains almost unchanged in the eastern coast and Bohai Bay. during the ®rst 10 model days, while the temperature For the same initial distribution of temperature, the near the bottom starts mixing up after 1 day (the model distributions of the temperature predicted by FVCOM boundary forcing is ramped up from zero to full am- and ECOM-si on the 10th model day are similar in the plitude over the ®rst 24 h of model integration) and interior but differ signi®cantly around the coast and is- reaches to an equilibrium state after 4 model days (Fig. lands. In the horizontal, both models predict a tidal mix- 10a). FVCOM shows relatively stronger mixing on the ing front around the 15-m isobath and a relatively uni- second model day, which probably is caused by the form temperature in the interior (Fig. 8). In the vertical, difference in horizontal resolution and water depth in- they also show the similar tidal mixing height above the terpolated from irregularly distributed dataset between bottom on sections 1 and 2 (Fig. 9). The major differ- these two models. At site 2 (around an island close to ence is that the cross-frontal gradient of temperature the Bohai Strait), the near-surface temperature decreases around the 15-m isobath is relatively larger in the case slightly with time in the case with ECOM-si, but drops with FVCOM than in the case with ECOM-si. Also, the more rapidly with time and also oscillates periodically model-predicted depth of the in section 1 after the fourth model day in the case with FVCOM is shallower in the case with FVCOM than in the case (Fig. 10b). Although temperature at middle depth and with ECOM-si. near the bottom predicted by these two models tends to Disparity in the ®eld of the temperature between mix up after 10 model days, the mixing rate seems faster FVCOM and ECOM-si is believed due to the difference in FVCOM than in ECOM-si (Fig. 10b). of the accuracy of geometric matching between these This is not a surprising result since the topography

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TABLE 2. Model±data comparison of the S2 tidal amplitude and phase in the Bohai Sea. Amplitude (cm) Phase (ЊG) FVCOM ECOM-si FVCOM ECOM-si

Stations ␨o ␨c ⌬␨ ␨c ⌬␨ ␾o ␾c ⌬␾ ␾c ⌬␾ 1 26.0 17.6 8.4 27.1 Ϫ1.1 348.0 349.7 Ϫ1.7 344.6 3.4 2 25.0 15.6 9.4 25.6 Ϫ0.6 355.0 358.7 Ϫ3.7 351.2 3.8 3 18.0 10.0 8.0 17.9 0.1 48.0 46.6 1.4 42.3 5.7 4 21.0 10.8 10.2 19.6 1.4 69.0 63.5 5.5 65.9 3.1 5 15.0 9.0 6.0 17.4 Ϫ2.4 139.0 144.5 Ϫ5.5 135.5 3.5 6 19.0 13.6 5.4 21.4 Ϫ2.4 167.0 166.5 0.5 156.3 10.7 7 37.0 25.9 11.1 38.1 Ϫ1.1 188.0 190.4 Ϫ2.4 180.5 7.5 8 32.0 26.1 5.9 36.8 Ϫ4.8 209.0 211.2 Ϫ2.2 203.7 5.3 9 27.0 20.7 6.3 27.6 Ϫ0.6 210.0 222.3 Ϫ12.3 212.5 Ϫ2.5 10 11.0 5.2 5.8 15.5 Ϫ4.5 216.0 243.9 Ϫ27.9 221.0 Ϫ5.0 11 5.0 2.6 2.4 3.7 1.3 205.0 263.7 Ϫ58.7 253.7 Ϫ48.7 12 2.0 1.5 0.5 2.3 Ϫ0.3 227.0 291.0 Ϫ64.0 300.7 Ϫ73.7 13 2.0 1.2 0.8 2.3 Ϫ0.3 203.0 310.5 Ϫ107.5 300.7 Ϫ97.7 14 1.0 3.0 Ϫ2.0 6.5 Ϫ5.5 39.0 4.1 34.9 18.0 21.0 15 5.0 4.4 0.6 8.6 Ϫ3.6 37.0 15.7 21.3 27.3 9.7 16 5.0 6.2 Ϫ1.2 10.8 Ϫ5.8 33.0 30.9 2.1 36.6 Ϫ3.6 17 11.0 7.8 3.2 10.8 0.2 65.0 51.4 13.6 36.6 28.4 18 20.0 17.6 2.4 20.4 Ϫ0.4 148.0 142.4 5.6 135.0 13.0 19 34.0 32.3 1.7 31.0 3.0 159.0 154.1 4.9 147.7 11.3 20 34.0 34.1 Ϫ0.1 31.0 3.0 165.0 161.3 3.7 147.7 17.3 21 30.0 36.4 Ϫ6.4 40.5 Ϫ10.5 177.0 174.2 2.8 182.2 Ϫ5.2 22 32.0 32.5 Ϫ0.5 40.5 Ϫ8.5 186.0 186.7 Ϫ0.7 182.2 3.8 23 24.0 21.8 2.3 22.4 1.6 196.0 194.5 1.5 203.2 Ϫ7.2 24 17.0 19.8 Ϫ2.8 10.8 6.2 45.0 23.0 22.0 204.1 Ϫ159.1 25 9.0 19.4 Ϫ10.4 30.2 Ϫ21.2 307.0 23.3 Ϫ76.3 57.7 Ϫ110.7 26 16.0 19.3 Ϫ3.3 27.7 Ϫ11.7 46.0 32.8 13.2 68.9 Ϫ22.9 27 18.0 16.9 1.1 20.4 Ϫ2.4 48.0 38.8 9.2 70.8 Ϫ22.8 28 13.0 10.3 2.7 9.6 3.4 38.0 35.0 3.0 54.6 Ϫ16.6 29 18.0 10.7 7.3 16.7 1.3 3.0 1.9 1.1 351.3 11.7 30 17.0 9.6 7.4 14.7 2.3 358.0 2.2 Ϫ4.2 351.0 7.0 31 21.0 14.0 7.0 10.4 10.6 351.0 345.4 5.6 338.5 12.5 32 22.0 15.9 6.1 26.3 Ϫ4.3 345.0 347.5 Ϫ2.5 336.3 8.7 Std dev 5.8 5.9 29.9 42.9

Note: ␨oÐobserved amplitude; ␨cÐcomputed amplitude; ⌬␨ ϭ ␨o Ϫ ␨c; ␾oÐobserved phase; ␾cÐcomputed phase; and ⌬␾ ϭ ␾o Ϫ ␾c. around the island is resolved well in FVCOM but not into 10 uniform ␴ layers, which correspond to a vertical in ECOM-si. If we believe that both FVCOM and resolution of less than 0.5 m in most areas inside the ECOM-si have the same numerical accuracy, then we river. The models were driven by the same semidiurnal could conclude here that poor matching of the complex M2 tidal forcing at the open boundary. The harmonic coastal geometries in the ®nite-difference model would constants of the M2 tidal forcing were speci®ed using underestimate mixing around the coast, which would the tidal elevations and phases predicted by the inner eventually lead to the unrealistic distribution of the tem- shelf South Atlantic Bight (SAB) tidal model [devel- perature in the interior, especially in a semienclosed oped and calibrated by Chen et al. (1999)]. No strati- coastal ocean like the Bohai Sea. Also, we learn from ®cation or river discharge is included in this model com- site 2 that the mismatch in the island geometry would parison experiment. ®lter a relatively large tidal oscillation near the surface, The model results show a signi®cant difference in the which tends to produce signi®cant mixing near the sur- along-river distribution of the M2 tidal amplitude be- face under conditions with no heat ¯ux. tween FVCOM and ECOM-si (Fig. 12). The observed

amplitude of the M2 tidal constituent is 94.7 Ϯ 1.3 cm at site 1, gradually increases to 99.4 Ϯ 1.4 cm at site b. The Satilla River 4, and then decreases to 96.0 Ϯ 1.3 cm at site 5. At The model grids of FVCOM and ECOM-si for the sites 6 and 7 in the southern and northern branches Satilla River are shown in Fig. 11. The horizontal res- separated at the upstream end of the main river channel, olution of ECOM-si is 100 m in the main channel of the observed amplitudes are 92.2 Ϯ 1.3 cm and 96.4 Ϯ the river and up to 2500 m near the open boundary in 1.3 cm, respectively. The amplitude of the sea level the inner shelf. Similar sizes of unstructured grids are predicted by ECOM-si increases upstream, with values used in FVCOM. In both models, the vertical is divided signi®cantly higher than the observed values at sites 4

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TABLE 3. Model±data comparison of the K1 tidal amplitude and phase in the Bohai Sea. Amplitude (cm) Phase (ЊG) FVCOM ECOM-si FVCOM ECOM-si

Stations ␨o ␨c ⌬␨ ␨c ⌬␨ ␾o ␾c ⌬␾ ␾c ⌬␾ 1 22.0 23.0 Ϫ1.0 19.1 2.9 7.0 0.9 6.1 5.4 1.6 2 22.0 20.5 1.5 18.6 3.4 14.0 10.8 3.2 11.9 2.1 3 24.0 22.9 1.1 20.3 3.7 57.0 54.9 2.1 49.6 7.4 4 30.0 26.0 4.0 23.7 6.3 68.0 60.5 7.5 58.1 9.9 5 33.0 32.2 0.8 28.4 4.6 81.0 75.9 5.1 74.2 6.8 6 39.0 34.9 4.1 30.2 8.8 95.0 79.0 16.0 78.4 16.6 7 43.0 40.2 2.8 35.2 7.8 85.0 84.4 0.6 83.2 1.8 8 38.0 40.6 Ϫ2.6 35.2 2.8 100.0 93.5 6.5 93.6 6.4 9 38.0 39.1 Ϫ1.1 33.2 4.8 104.0 98.1 5.9 98.0 6.0 10 35.0 33.3 1.7 30.2 4.8 107.0 102.3 4.7 101.1 5.9 11 33.0 32.3 0.7 26.9 6.1 91.0 103.7 Ϫ12.7 103.8 Ϫ12.8 12 37.0 31.7 5.3 26.2 10.8 107.0 105.1 1.9 105.1 1.9 13 32.0 31.3 0.7 26.2 5.8 95.0 105.8 Ϫ10.8 105.1 Ϫ10.1 14 29.0 30.6 Ϫ1.6 25.2 3.8 108.0 109.9 Ϫ1.9 110.1 Ϫ2.1 15 29.0 30.7 Ϫ1.7 25.0 4.0 111.0 112.2 Ϫ1.2 112.9 Ϫ1.9 16 28.0 30.5 Ϫ2.5 24.9 3.1 114.0 116.3 Ϫ2.3 117.1 Ϫ3.1 17 28.0 29.7 Ϫ1.7 24.9 3.1 127.0 122.7 4.3 117.1 9.9 18 31.0 34.2 Ϫ3.2 26.4 4.6 153.0 145.8 7.2 147.0 6.0 19 42.0 39.1 2.9 29.4 12.6 154.0 146.9 7.1 147.4 6.6 20 36.0 40.0 Ϫ4.0 29.4 6.6 154.0 149.6 4.4 147.4 6.6 21 28.0 40.9 Ϫ12.9 32.1 Ϫ4.1 140.0 154.8 Ϫ14.8 159.9 Ϫ19.9 22 38.0 40.3 Ϫ2.3 32.1 5.9 161.0 160.3 0.7 159.9 1.1 23 29.0 37.3 Ϫ8.3 28.8 0.2 174.0 164.4 9.6 170.2 3.8 24 27.0 32.1 Ϫ5.1 25.8 1.2 197.0 187.3 9.7 171.1 25.9 25 18.0 31.8 Ϫ13.8 23.0 Ϫ5.0 180.0 187.5 Ϫ7.5 196.0 Ϫ16.0 26 25.0 31.5 Ϫ6.5 22.5 2.5 194.0 193.2 0.8 202.8 Ϫ8.8 27 22.0 30.4 Ϫ8.4 20.6 1.4 197.0 197.1 Ϫ0.1 206.7 Ϫ9.7 28 20.0 25.5 Ϫ5.5 17.3 2.7 200.0 203.9 Ϫ3.9 214.5 Ϫ14.5 29 7.0 3.6 3.4 3.7 3.3 17.0 338.3 38.7 345.2 31.8 30 1.0 6.7 Ϫ5.7 5.2 Ϫ4.2 179.0 242.7 Ϫ63.7 258.5 Ϫ79.5 31 15.0 18.4 Ϫ3.4 14.1 0.9 281.0 277.4 3.6 239.2 41.8 32 16.0 21.3 Ϫ5.3 16.8 Ϫ0.8 295.0 295.3 Ϫ0.3 288.5 6.5 Std dev 5.1 5.2 14.8 19.3

Note: ␨oÐobserved amplitude; ␨cÐcomputed amplitude; ⌬␨ ϭ ␨o Ϫ ␨c; ␾oÐobserved phase; ␾cÐcomputed phase; and ⌬␾ ϭ ␾o Ϫ ␾c. and 5. Since ECOM-si fails to resolve the two river sites, Zheng et al. (2002b) tuned the model by adjusting branches at the upstream end of the main channel, water the bottom roughness zo. Since we have not yet added ¯ooding up the river tends to accumulate there. In con- ¯ooding/drying to FVCOM, the model comparison trast, FVCOM not only predicts the same trend of the made is between both models without this process.

M2 tidal amplitude as the observations from site 4 to Tidal currents computed by FVCOM and ECOM-si 7, but also their values agree with each other within also differ signi®cantly, especially around the estuary± measurement uncertainty. In addition, FVCOM shows tidal creek area (Figs. 13 and 14). FVCOM shows a higher values of the amplitude than the observations at relatively strong tidal current near both the southern and sites 1±3, which is believed due to the ¯ooding/drying northern coasts, with a substantial in¯ow to and out¯ow process over the intertidal zone around the mouth of the from tidal creeks during ¯ood and ebb tides, respec- river. tively. These patterns are not resolved in ECOM-si. Zheng et al. (2002b) incorporated a 3D wet/dry point FVCOM predicts a stronger along-coast residual ¯ow treatment method into ECOM-si and used it to simulate near the tidal creek, which intensi®es the topographi- cally induced eddylike residual circulation cell on the the amplitude and phase of the M2 tidal constituent in the Satilla River. They found that the ¯ooding/drying eastern side of the tidal creek (Fig. 15a). Although this process plays a key role in simulating tidal elevation eddylike residual circulation cell is also predicted in and currents in the main river channel. Including the ECOM-si, it is much weaker and the velocity is sym- intertidal zone in the ECOM-si did show a signi®cant metrically distributed relative to its center (Fig. 15b). improvement in the simulation of tidal elevation at site 5, but it still fails to provide reasonable values of the 6. Discussion and summary amplitude at sites 6 and 7. To make the model-predicted An unstructured grid, ®nite-volume, three-dimen- tidal elevation match the observed value at measurement sional primitive equation coastal ocean model

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TABLE 4. Model±data comparison of the O1 tidal amplitude and phase in the Bohai Sea. Amplitude (cm) Phase (ЊG) FVCOM ECOM-si FVCOM ECOM-si

Stations ␨o ␨c ⌬␨ ␨c ⌬␨ ␾o ␾c ⌬␾ ␾c ⌬␾ 1 17.0 15.2 1.8 16.2 0.8 328.0 319.0 9.0 327.8 0.2 2 16.0 13.7 2.3 16.0 0.0 334.0 331.7 2.3 334.9 Ϫ0.9 3 19.0 17.7 1.3 18.9 0.1 10.0 13.9 Ϫ3.9 12.9 Ϫ2.9 4 23.0 20.0 3.0 22.0 1.0 17.0 17.9 Ϫ0.9 20.4 Ϫ3.4 5 25.0 24.6 0.4 26.4 Ϫ1.4 32.0 30.3 1.7 35.0 Ϫ3.0 6 27.0 26.4 0.6 28.1 Ϫ1.1 45.0 32.8 12.2 38.7 6.3 7 30.0 29.8 0.2 32.2 Ϫ2.2 46.0 37.3 8.7 42.9 3.1 8 26.0 30.0 Ϫ4.0 32.8 Ϫ6.8 47.0 45.6 1.4 52.5 Ϫ5.5 9 29.0 29.1 Ϫ0.1 31.4 Ϫ2.4 54.0 49.9 4.1 56.5 Ϫ2.5 10 26.0 25.6 0.4 29.2 Ϫ3.2 50.0 53.9 Ϫ3.9 59.2 Ϫ9.2 11 25.0 25.0 0.0 26.7 Ϫ1.7 53.0 55.2 Ϫ2.2 61.5 Ϫ8.5 12 26.0 24.6 1.4 26.2 Ϫ0.2 52.0 56.3 Ϫ4.3 62.6 Ϫ10.6 13 23.0 24.4 Ϫ1.4 26.2 Ϫ3.2 52.0 57.0 Ϫ5.0 62.6 Ϫ10.6 14 27.0 24.1 2.9 25.6 1.4 61.0 60.5 0.5 66.6 Ϫ5.6 15 23.0 24.2 Ϫ1.2 25.7 Ϫ2.7 61.0 62.3 Ϫ1.3 68.9 Ϫ7.9 16 25.0 24.2 0.8 25.7 Ϫ0.7 67.0 65.8 1.2 72.2 Ϫ5.2 17 24.0 23.8 0.2 25.7 Ϫ1.7 69.0 71.2 Ϫ2.2 72.2 Ϫ3.2 18 25.0 27.1 Ϫ2.1 27.8 Ϫ2.8 94.0 89.9 4.1 95.4 Ϫ1.4 19 30.0 30.5 Ϫ0.5 30.5 Ϫ0.5 96.0 90.4 5.6 95.5 0.5 20 27.0 31.1 Ϫ4.1 30.5 Ϫ3.5 99.0 92.8 6.2 95.5 3.5 21 18.0 31.7 Ϫ13.7 32.7 Ϫ14.7 103.0 97.4 5.6 105.8 Ϫ2.8 22 31.0 31.4 Ϫ0.4 32.7 Ϫ1.7 111.0 102.6 8.4 105.8 5.2 23 27.0 29.4 Ϫ2.4 29.6 Ϫ2.6 116.0 106.7 9.3 114.5 1.5 24 24.0 26.0 Ϫ2.0 26.9 Ϫ2.9 132.0 126.6 5.4 115.4 16.6 25 12.0 25.8 Ϫ13.8 23.6 Ϫ11.6 132.0 126.9 5.1 136.0 Ϫ4.0 26 26.0 25.6 0.4 22.9 3.1 138.0 132.2 5.8 142.2 Ϫ4.2 27 23.0 24.8 Ϫ1.8 21.2 1.8 132.0 135.8 Ϫ3.8 145.8 Ϫ13.8 28 17.0 21.2 Ϫ4.2 17.8 Ϫ0.8 138.0 141.8 Ϫ3.8 152.7 Ϫ14.7 29 4.0 1.3 2.7 1.7 2.3 12.0 68.1 Ϫ56.1 4.3 7.7 30 3.0 6.0 Ϫ3.0 4.3 Ϫ1.3 63.0 151.7 Ϫ88.7 170.8 Ϫ107.8 31 9.0 12.3 Ϫ3.3 13.5 Ϫ4.5 232.0 206.2 25.8 175.5 56.5 32 9.0 13.4 Ϫ4.4 14.0 Ϫ5.0 234.0 225.7 8.3 240.5 Ϫ6.5 Std dev 4.0 4.1 19.8 22.6

Note: ␨oÐobserved amplitude; ␨cÐcomputed amplitude; ⌬␨ ϭ ␨o Ϫ ␨c; ␾oÐobserved phase; ␾cÐcomputed phase; and ⌬␾ ϭ ␾o Ϫ ␾c.

(FVCOM) has been developed for the study of coastal circular lake, and tidal wave resonance in a simple sem- and estuarine circulation. This model combines the ad- ienclosed channel. The results show in the ®rst case that vantages of the ®nite-element method for geometric poor resolution of the curved coastal geometry causes ¯exibility and ®nite-difference method for simple dis- both unwanted wave damping and a time-dependent crete computational ef®ciency. The numerical experi- phase shift. In the second case, the near-resonance be- ments in the Bohai Sea and Satilla River demonstrate havior is strongly in¯uenced by channel shape irregu- that this model provides a more accurate simulation of laries. This may explain why FVCOM provides a more tidal currents and residual ¯ow in coastal ocean and accurate simulation for the amplitude of the M2 tidal estuarine settings where multiple islands, inlets, and tid- constituent in Bohai Bay. A manuscript describing these al creeks exist. Because of a better ®tting of the geo- and other idealized model comparisons is in preparation. metric complex in FVCOM, this model should provide The goal of this paper is to introduce the unstructured a more accurate representation of water mass property grid, ®nite-volume numerical approach to the coastal variability and the advection and mixing of passive trac- ocean community. We fully understand that more ex- ers around the coast. periments and comparisons with analytical solutions and FVCOM and ECOM-si show similar accuracy in the other models must be made in order to validate the tracer simulation experiments except around complex usefulness and reliability of this new ®nite-volume topographies. Regarding the ®nite-difference approach, ocean model for the study of coastal and estuarine cir- the most signi®cant improvement provided by FVCOM culation and ecosystem dynamics. is the geometric ¯exibility with unstructured grids. Re- Recently, a wet/dry point treatment technique was cently, some model experiments were conducted with introduced into FVCOM. It is now being tested in the FVCOM, ECOM-si, and POM for two idealized cases Satilla River, an estuary characterized by intensive in- with analytic solutions: free long gravity waves in a tertidal salt marshes. Also, a Lagrangian particle track-

Unauthenticated | Downloaded 09/27/21 06:45 PM UTC JANUARY 2003 CHEN ET AL. 179 ing code was added into the FVCOM code, and is being tested through comparison with ECOM-si. Water quality and suspended sediment models are also being devel- oped. The formulations of these models are the same as the water quality and suspended sediment models we developed for Georgia estuaries based on ECOM-si (Zheng et al. 2002a). Hopefully, as FVCOM matures, others will join in our efforts to make this an important tool to better understand our coastal environment.

Acknowledgments. This research was supported by the Georgia Sea Grant College Program under Grants NA26RG0373 and NA66RG0282, the National Science Foundation under Grant OCE-97-12869, and the U.S. GLOBEC Georges Bank Program through support from NOAA Grants NA56RG0487, NA96OP003, and NA96OP005 and NSF Grant OCE 98-06379. We want to especially thank M. Rawson, manager of the Georgia Sea Grant College Program, for his strong encourage- ment and support of this model development effort. Without the continuous ®nancial support of the Georgia Sea Grant College Program, this model development FIG. A1. The illustration of the local coordinates. would not have been possible. We also want to thank A. Blumberg, who gave us permission to use ECOM- 3 si during the last 10 years. Our experience with ECOM- ADVU ϭ (uD´ ␷ lˆ ), ͸ im m nm m si has helped us build a solid foundation in modeling, mϭ1 which in turn has directly bene®ted us in developing 3 FVCOM. G. Davidson (Georgia Sea Grant College Pro- ADVV ϭ (␷ D ´ ␷ lˆ ), (A.3) ͸ im m nm m gram) provided editorial help on this manuscript. His mϭ1 assistance is greatly appreciated. Three anonymous re- where u , ␷␷, and are the x, y, and normal com- viewers provided many critical comments and construc- im im nm ponents of the velocity on the side line m of a triangle tive suggestions, which really helped us to improve the cell, and ␷ is positive when its direction is outward. model±data comparison and clarify the ®nal manuscript. nm Here ˆlm and D m are the length and midpoint water depth APPENDIX of the side line m, respectively. They are equal to

Dmiϭ 0.5[D(N ( j1)) ϩ D(Ni( j2))] (A.4) The Discrete Form of the 2D External and 2 3D Internal Modes lˆmniϭ {[X (N ( j1)) Ϫ Xni(N ( j2))]

21/2 a. The 2D external mode ϩ [Yni(N ( j1) Ϫ Yni(N ( j2))] } , (A.5)

nnwhere In the 2D external mode equationsRRu and␷ are expressed by m ϩ 1 Rn ϭ ADVU ϩ DPBPX ϩ DPBCX ϩ CORX j2 ϭ m ϩ 1 Ϫ INT ϫ 3; u ΂΃4 ϩ VISCX Ϫ G , (A.1) x m ϩ 2 n j1 ϭ m ϩ 2 Ϫ INT ϫ 3. (A.6) R␷ ϭ ADVV ϩ DPBPY ϩ DPBCY ϩ CORY ΂΃4

ϩ VISCY Ϫ Gy, (A.2) The velocity in the triangle cell i is assumed to satisfy the linear distribution given as where ADVU and ADVV, DPBPX and DPBPY, DPBCX uuu and DPBCY, CORX and CORY, VISCX and VISCY are uiii,(xЈ, yЈ) ϭ ␾ (xЈ, yЈ) ϭ u 0 ϩ axiiЈϩbyЈ, (A.7) the x and y components of vertically integrated hori- (x , y ) ␷␷␷(x , y ) ax by, (A.8) zontal advection, barotropic pressure gradient force, ␷ iii,Ј Ј ϭ ␾ Ј Ј ϭ ␷ 0 ϩ iiЈϩ Ј uu␷␷ Coriolis force, and horizontal diffusion terms, respec- where the parametersabaiii , , , and b i are determined tively. The de®nitions of Gx and Gy are the same as by a least squares method based on velocity values at those shown in Eqs. (2.40) and (2.41) in the text. the four cell centered points shown in Fig. A1 (one cal- The x and y components of the horizontal advection culated cell plus three surrounding cells). Then, the nor- are calculated numerically by mal velocity component on the side line m is given as

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␷ nmϭ ␷à mcos␪ Ϫ uà m sin␪, (A.9) DPBCX 00␳ץץ where g ϭ DD␳ d␴Ј d␴ dx dy ␴Јץ ͵ ͵ xץ͵͵ ␳o Y (N ( j )) Ϫ Y (N ( j )) Ά []Ϫ1 ΂΃␴ ␪ ϭ arctanni2 nii , and (A.10) 0 ץ (( Xni(N ( j2)) Ϫ Xnii(N ( j 2 uà 0.5[ uu(x , y ) (x , y )], ϩ D ␳␴ d␴ dx dy ͵ xץ͵͵ imϭ ␾ iЈ mЈ mϩ ␾ NBi(m) ЈmmЈ ΂΃Ϫ1 · ␷␷ ␷à imϭ 0.5[␾ i(x Ј m, y Ј m) ϩ ␾ NBi(m)(x Јmm, y Ј )], (A.11) ␳ץg 00 ϭ DD ␴ d␴Ј d␴ dy wherexЈmm andyЈ are the midpoint of the side line m. ␴Јץ͵͵␳o Ͷ Ά []Ϫ1 ΂΃␴ The momentum ¯ux through three side sections of triangle cell i is calculated using a second-order accu- 0 2 racy (Kobayashi et al. 1999; Hubbard 1999) as follows: ϩ D Ͷ͵␳␴ d␴ dy . (A.16) ΂΃Ϫ1 · The discrete form of Eq. (A.16) is given as ␾ u(0, 0), ␷ Ͻ 0 u ϭ inm im ␾ u (x , y ), ␷ Ն 0, 0.5g 3 Ά NBi(m) im im nm DPBCX ϭ DD[PB (i) ϩ PB (NB (m))] im͸ 12i ␳o Ά mϭ1 ␾ ␷ (0, 00, ␷ Ͻ 0 ␷ ϭ inm(A.12) ϫ [Y (N ( j )) Ϫ Y (N ( j ))] im ␷ ni1 ni2 Ά␾ NBi(m)(xim, y im), ␷ nm Ն 0, 3 2 ϩ D [PB (i) Ϫ PB (NB (m))] where xim and yim are the cell-centered point of the sur- i ͸ 22i mϭ1 rounding triangle numbered NBi(m), and (0, 0) indicates the location of the cell-centered point. The area integration of barotropic pressure gradient ϫ [Yni(N ( j1)) Ϫ Yni(N ( j2))] , (A.17) force terms can be converted to a trajectory integration · using Stokes' theorem. They can then be calculated nu- where merically by a simple discrete method as follows: KBϪ1 PB (i) ϭ [␴(kЈ) Ϫ ␴(kЈϩ1)] DPBPX 1 ͸ kЈϭ1 Ά 3

ϭ gD ␨ [Y (N ( j )) Ϫ Y (N ( j ))], (A.13) kЈ imni͸ 1 ni2 mϭ1 ϫ ͸ ␴(k)[␳(k) Ϫ ␳(k ϩ 1)] (A.18) DPBPY kϭ1 · 3 KBϪ1 PB (i) ϭ 0.5 [␳(k) ϩ ␳(k ϩ 1)] ϭ gDimni␨ [X (N ( j2)) Ϫ Xni(N ( j1))], (A.14) 2 ͸ ͸ kЈϭ1 mϭ1 ϫ ␴(k)[␴(k) Ϫ ␴(k ϩ 1)]. (A.19) where ␨ m ϭ 0.5[␨(Ni(j1)) ϩ ␨(Ni(j2))]. A similar approach is used to calculate the baroclinic Similarly, we can derive the y component of the baro- pressure gradient force terms. These terms are rewritten clinic pressure gradient force as into the form of the gradient to take the advantage of 0.5g 3 the ¯ux calculation in the ®nite-volume method. For DPBCY ϭ DD[PB (i) ϩ PB (NB (m))] im͸ 12i example, the x component of the baroclinic pressure ␳ 0 Ά mϭ1 gradient force can be rewritten as ϫ [Xni(N ( j2)) Ϫ Xni(N ( j1))] 0 3 D 2ץץ gD D d ϩ D [PB (i) ϩ PB (NB (m))] Ϫ ␳ ␴Јϩ␴␳ i ͸ 22i x mϭ1ץ ͵ xץ ␳o []΂΃␴

0 (␳␴ ϫ [X (N ( j )) Ϫ X (N ( j ))] . (A.20ץץ gD ϭϪ D ␳ d␴ ϩ ␴␳D Ϫ D ni2 ni1 · xץ ͵ xץ ␳o Ά·[]␴ The discrete forms of the Coriolis force terms are ␳␴ given asץ ␳ 0ץץ gD ϭ D ␴ d␴ ϩ D . (A.15) x u ␷ץ ␴ץ ͵ xץ ␳o Ά·␴ CORX f D ; CORY fu D . (A.21) ϭϪ ␷ iii⍀ ϭ iii⍀ Integrating Eq. (A.15) from Ϫ1 to 0 and then integrating The x and y components of the horizontal diffusion over a triangle cell area again, we get can be rewritten as

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␷ץ uץץ uץץ DFÄ dx dy ഠ 2AH ϩ AH ϩ dx dy · xץ yץy[] ΂΃ץ xץx΂΃mmץx ͵͵ Ά ͵͵ ␷ץ uץ uץ ϭ 2 AH dyϪ AH ϩ dx and (A.22) xץ yץx Ͷ m ΂΃ץ Ͷ m ␷ץ uץץ ␷ץץ DFÄ dx dy ഠ 2AH ϩ AH ϩ dx dy · xץ yץx[] ΂΃ץ yץy΂΃mmץy ͵͵ Ά ͵͵ ␷ץ uץ ␷ץ ϭϪ2 AH dxϩ AH ϩ dy. (A.23) xץ yץy Ͷ m ΂΃ץ Ͷ m The discrete forms of Eqs. (A.22) and (A.23) are given as

3 VISCX ϭ 0.5H [A (i) ϩ A (NB(m))][auu(i) ϩ a (NB(m))][Y (N ( j )) Ϫ Y (N ( j ))] ͸ mm m ni1 ni2 mϭ1 Ά uu ␷␷ ϩ 0.25Hmm[A (i) ϩ A m(NB(m))][b (i) ϩ b (NB(m)) ϩ a (i) ϩ a (NB(m))]

[X (N ( j )) Ϫ X (N ( j ))] , (A.24) ni2 ni1 · where H m ϭ 0.5[H(Ni(j1)) ϩ H(Ni(j 2))], and

3 VISCY ϭ 0.5H [A (i) ϩ A (NB(m))][b␷␷(i) ϩ b (NB(m))][X (N ( j )) Ϫ X (N ( j ))] ͸ mm m ni2 ni1 mϭ1 Ά uu ␷␷ ϩ 0.25Hmm[A (i) ϩ A m(NB(m))][b (i) ϩ b (NB(m)) ϩ a (i) ϩ a (NB(m))]

ϫ [Y (N ( j )) Ϫ Y (N ( j ))] . (A.25) ni1 ni2 ·

The Gx and Gy are given as

Gx ϭ ADVU ϩ VICX Ϫ ADVU Ϫ VISCX (A.26)

Gy ϭ ADVV ϩ VICY Ϫ ADVV Ϫ VISCY, (A.27) where u␷ Dץ uD2ץ ADVU ϭϩdx dy ϭ uDdy2 ϩ u␷ Ddx y Ͷ Ͷץ xץ[]͵͵ 3 ϭ 0.5{[u22(i) ϩ u (NB(m))]D [Y (N ( j )) Ϫ Y (N ( j ))] ͸ mn i1 ni2 mϭ1

ϩ [u(i)␷(i) ϩ u(NB(m))␷(NB(m))]Dmni[X (N ( j2)) Ϫ Xni(N ( j1))]}; (A.28)

␷ 2Dץ u␷ Dץ ADVV ϭϪ ϩ dx dy ϭϪ u␷ DdyϪ ␷ 2Ddx y Ͷ Ͷץ xץ[]͵͵ 3 ϭ 0.5{[u(i)␷(i) ϩ u(NB(m))␷(NB(m))]D [Y (N ( j )) Ϫ Y (N ( j ))] ͸ mn i1 ni2 mϭ1

22 ϩ [␷ (i) ϩ ␷ (NB(m))]Dmni[X (N ( j2)) Ϫ Xni(N ( j1))]}; (A.29)

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␷ץ uץץ uץץ VISCX ϭ DF dx dy ഠ 2AH ϩ AH ϩ dx dy xץ yץy ΂΃ץ xץx mmץ[] ͵͵ x ͵͵ ␷ץ uץ uץ ␷ץ uץ uץ ϭ 2AH dyϪ AH ϩ dx ϭ 2 HA dyϪ HA ϩ 2 dx; (A.30) xץ yץx Ͷ []m ΂΃ץ x Ͷ ΂΃mץ yץx Ͷ []m ΂΃ץ Ͷ ΂΃m and

␷ץ uץץ ␷ץץ VISCY ϭ DF dx dy ഠ 2AH ϩ AH ϩ dx dy xץ yץx ΂΃ץ yץy mmץ[] ͵͵ y ͵͵ ␷ץ uץ ␷ץ ϭϪ 2AH dxϩ AH ϩ dy xץ yץy Ͷ []m ΂΃ץ Ͷ ΂΃m ␷ץ uץ ␷ץ ϭϪ2 HA dxϩ HA ϩ dy. (A.31) x ΃ץ yץy Ͷ []m ΂ץ Ͷ ΂΃m Let us de®ne

␷ץ ␷ץ uץ uץ USH ϭ A , UVSH ϭ A ϩ , and VSH ϭ A , (A.32) yץx mץ yץx ΂΃ץmm where u and ␷ are the x and y components of the velocity output from the 3D model. At each ␴ level in a triangle cell, they can be expressed as a linear function as

uu ␷␷ ui,k(xЈ, yЈ) ϭ u i,k(0, 0) ϩ ax(i,k)(Јϩbyi,k) Ј, ␷ i,k(xЈ, yЈ) ϭ ␷ i,k(0, 0) ϩ ax(i,k)(Јϩbyi,k) Ј. (A.33) Then at the triangle cell i, we have u KBϪ1ץ USH(i) ϭ A ϭ A (k)au , (A.34) mm͸ (i,k) x kϭ1ץ ␷ KBϪ1ץ VSH(i) ϭ A ϭ A (k)b␷ , (A.35) mm͸ (i,k) y kϭ1ץ

␷ KBϪ1ץ uץ u ␷ UVSH(i) ϭ AmmϩϭA (k)[a(i,k)(ϩ b i,k)]. (A.36) x k͸ϭ1ץ yץ΂΃ Therefore,

␷ץ uץ uץ VISCX ϭ 2 HA dyϪ H A ϩ dx x ΃ץ yץx Ͷ []m ΂ץ Ͷ ΂΃m 3 ϭ H [USH(i) ϩ USH(NB(m))][Y (N ( j )) Ϫ Y (N ( j ))] ͸ mni1 ni2 mϭ1

3 ϩ 0.5 H [UVSH(i) ϩ UVSH(NB(m))][X (N ( j )) Ϫ X (N ( j ))]; (A.37) ͸ mni2 ni1 mϭ1

␷ץ uץ ␷ץ VISCY ϭϪ2 HA dxϩ HA ϩ 2 dy x ΃ץ yץy Ͷ []m ΂ץ Ͷ ΂΃m 3 ϭ H [VSH(i) ϩ VSH(NB(m))][X (N ( j )) Ϫ X (N ( j ))] ͸ mni2 ni1 mϭ1

3 ϩ 0.5 H [UVSH(i) ϩ UVSH(NB(m))][Y (N ( j )) Ϫ Y (N ( j ))]. (A.38) ͸ mni1 ni2 mϭ1

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b. The 3D internal mode

The 3D momentum equations can be rewritten as

␷ץץ ␷D 1ץ uץץ uD 1ץ ϩ R ϭ K , ϩ R ϭ K , (A.39) ␴ץ ␴ ΂΃mץ tD␷ץ ␴ץ␴΂΃ץtDumץ

where

Ru ϭ ADVU3 ϩ CORX3 ϩ HVISCX ϩ DPBPX3 ϩ BPBCX3, (A.40)

R␷ ϭ ADVV3 ϩ CORY3 ϩ HVISCY ϩ DPBPY3 ϩ BPBCY3. (A.41)

The numerical integration is conducted in two steps. At the ®rst step, the ``transition'' velocity is calculated using all the terms except the vertical diffusion term in the momentum equations. Then the true velocity is determined implicitly using a balance between the local change of the ``transition'' velocity and the vertical diffusion term.

Letu*i,k and␷* i,k be the x and y components of the ``transition'' velocity at the midpoint between k and k ϩ 1 ␴ levels in triangle cell i. They can be determined numerically as follows:

⌬t ⌬t n IInnn u*i,kϭ u i,kϪ R u,(i,k), ␷*i,kϭ ␷ i,k Ϫ R␷,(i,k), (A.42) ⍀⌬ii␴D ⍀⌬ ii␴D

where ⌬␴ ϭ ␴k Ϫ ␴kϩ1, ⌬tI is the time step for the internal mode. nn Each term inRRu,(i,k) and␷,(i,k) is computed as follows: u␻ץ u␷Dץ uD2ץ ␴k ADVU3n ϭϩϩd␴ dx dy ␴ץ yץ xץ ͵ ͵͵ (i,k) ␴ ΂΃ []kϩ1

3 ϭ (␴ Ϫ ␴ ) unn(m)D ␷ (m)lˆ ϩ⍀[(u nnnnnnϩ u )␻ Ϫ (u ϩ u )␻ ]; (A.43) kkϩ1 ͸ i,k m n,k m i i,kϪ1 i,k i,k i,k i,kϩ1 i,kϩ1 mϭ1

␷␻ץ ␷ 2Dץ u␷Dץ ␴k ADVV3n ϭϩϩd␴ dx dy ␴ץ yץ xץ ͵ ͵͵ (i,k) ␴ ΂΃ []kϩ1

3 ϭ (␴ Ϫ ␴ ) ␷ nn(m)D ␷ (m)lˆ ϩ⍀[(␷ nnnnnnϩ ␷ )␻ Ϫ (␷ ϩ ␷ )␻ ]; (A.44) kkϩ1 ͸ i,k m n,k m i i,kϪ1 i,k i,k i,k i,kϩ1 i,kϩ1 mϭ1

CORX3 ϭϪf ␷ iiD (␴ kϪ ␴ kϩ1)⍀iiikk, CORY3 ϭ fu D (␴ Ϫ ␴ ϩ1)⍀i; (A.45)

␷ץ uץץ uץץ00 HVISCX n ϭ DF d␴ dx dy ഠ 2AH ϩ AH ϩ d␴ dx dy xץ yץ yץ xץx mmץ ͵ ͵͵ i,k) ͵͵ ͵ x) ΂΃␴␴Ά [][΂΃ ]·

␷ץ uץ uץ ϭ 2 AH dyϪ AH ϩ dx (␴ Ϫ ␴ ) x ϩ1ץ yץx Ͷ mkk΂΃ץ Ͷ m[] 3 ϭ 0.5H [A (i) ϩ A (NB(m))](auuϩ a ][Y (N ( j )) Ϫ Y (N ( j ))] ͸ mm m (i,k)(NB(m),k) ni1 ni2 Άmϭ1 3 ϩ 0.25H (buuϩ b ϩ a␷␷ϩ a ][X (N ( j )) Ϫ X (N ( j ))] ͸ m (i,k)(NB(m),k)(i,k)(NB(m),k) ni2 ni1 mϭ1

ϫ [A (i) ϩ A (NB(m))] (␴ Ϫ ␴ ); (A.46) mm· kkϩ1

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␷ץ uץץ ␷ץץ00 HVISCY n ϭ DF d␴) dx dy ഠ 2AH ϩ AH ϩ d␴ dx dy xץ yץ xץ yץy mmץ ͵ ͵͵ i,k) ͵͵ ͵ y) ΂ ␴␴Ά [][΂΃ ]·

␷ץ uץ ␷ץ ϭϪ2 AH dxϩ AH ϩ dy (␴ Ϫ ␴ ) x ϩ1ץ yץy Ͷ mkk΂΃ץ Ͷ m[]

3 ϭ 0.5H [A (i) ϩ A (NB(m))](b␷␷ϩ b ][X (N ( j )) Ϫ X (N ( j ))] ͸ mm m (i,k)(NB(m),k) ni2 ni1 Άmϭ1

3 ϩ 0.25H (buuϩ b ϩ a␷␷ϩ a ][Y (N ( j )) Ϫ Y (N ( j ))] ͸ m (i,k)(NB(m),k)(i,k)(NB(m),k) ni1 ni2 mϭ1

ϫ [A (i) ϩ A (NB(m))] (␴ Ϫ ␴ ); (A.47) mm· kkϩ1

3 n DPBPX n ϭ gD (␴ Ϫ ␴ ) ␨ [Y (N ( j )) Ϫ Y (N ( j ))]; (A.48) (i,k) ik kϩ1 ͸ mn i1 ni2 mϭ1

3 n DPBPY n ϭ gD (␴ Ϫ ␴ ) ␨ [X (N ( j )) Ϫ X (N ( j ))] (A.49) (i,k) ik kϩ1 ͸ mni2 ni1 mϭ1

␳␴ץ ␳ ␴k 0ץץ g ␴k PBCX3 ϭϪ DD ␴ d␴ d␴ dx dy ϩ Dd2 ␴ dx dy xץ ͵͵͵ ␴ץ͵ xץ͵͵͵ ␳o ␴␴΂΃ ␴ Ά kϩ1 [] kϩ1 ·

␳ץ g 0 2 ϭϪ DDi ␴ d␴ dy ϩ Di ␳␴ dy [␴kkϪ ␴ ϩ1]. (A.50) ␴ Ͷץ͵␳o Ͷ Ά ΂΃␴ · Let ␳ kץ 0 PBC(i) ϭ ␴ d␴ ϭ ͸ ␴(kЈ)[␳(kЈ) Ϫ ␳(kЈϩ1)], (A.51) ␴ kЈϭ1ץ ͵ ␴ then

0.5g 3 DPBCX ϭϪ (␴ Ϫ ␴ ) DD[PBC(i) ϩ PBC(NB (m))][Y (N ( j )) Ϫ Y (N ( j ))] kkϩ1 im͸ i ni1 ni2 ␳o Ά mϭ1

3 ϩ D 2 [␳(i) ϩ ␳(NB (m))]␴(k)[Y (N ( j )) Ϫ Y (N ( j ))] . (A.52) iini͸ 1 ni2 mϭ1 · Similarly, we can derive the y component of the baroclinic pressure gradient as

0.5g 3 DPBCY ϭϪ (␴ Ϫ ␴ ) DD[PBC(i) ϩ PBC(NB (m))][X (N ( j )) Ϫ X (N ( j ))] kkϩ1 im͸ i ni2 ni1 ␳o Ά mϭ1

3 ϩ D 2 [␳(i) ϩ ␳(NB (m))]␴(k)[X (N ( j )) Ϫ X (N ( j ))] . (A.53) iini͸ 2 ni1 mϭ1 · The mathematic forms of the two equations in (A.39) are the same, so that they can be solved numerically using the same approach. The method used to numerically solve these equations was adopted directly from the ECOM-si (Blumberg 1994). For example, a detailed description of this method is given below for the u component of the momentum equation. The implicit discrete form of the ®rst equation in (A.39) is given as

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nϩ1 nϩ1 nϩ1 Aui,k kϩ1 ϩ Bui,k kϩ Cu i,k kϪ1 ϭ u*, (A.54) where 2K (k ϩ 1)⌬t 2K (k)⌬t A ϭϪ m ; C ϭϪ m ; i,knϩ12 i,k nϩ12 [D ](␴kkϪ ␴ ϩ1)(␴kkϪ ␴ ϩ1)[D ](␴kkϪ ␴ ϩ1)(␴kϪ1 Ϫ ␴kϩ1)

Bi,kϭ 1 Ϫ A i,kϪ C i,k. (A.55) This is a tridiagonal equation and it ranges from k ϭ 2 to KB-2, where KB is the number of total ␴ levels in the vertical. The solution for u(k) is calculated by Au* Ϫ CVHP(k Ϫ 1) u(k) ϭϪ i,ku(k ϩ 1) ϩ i,k , (A.56) Bi,kϩ CVH i,k(k Ϫ 1) B i,kϩ CVH i,k (k Ϫ 1) where Au* Ϫ CVHP(k Ϫ 1) VH(k) ϭϪ i,k; VHP(k) ϭ i,k i,k . (A.57) Bi,kϩ CVH i,k(k Ϫ 1) B i,kϩ CVH i,k (k Ϫ 1) The equation for temperature or salinity as well as other passive tracers also can be rewritten as the form shown in (A.39), so they can be solved numerically using the exact same approach discussed above. The only difference is that ␪ is calculated at nodes and has the same control volume as that used for ␨. To shorten the text, we decide not to include the detailed description of the ®nite-volume numerical approach for tracer equations here.

REFERENCES for the South Atlantic Bight. Estuarine Processes, Vol. 1, M. Wiley, Ed., Academic Press, 69±78. Galperin, B., L. H. Kantha, S. Hassid, and A. Rosati, 1988: A quasi- Barth, T. J., 1993: Recent developments in high order K-exact re- equilibrium turbulent energy model for geophysical ¯ows. J. construction on unstructured meshes. AIAA Paper 93±0668. Atmos. Sci., 45, 55±62. Bigham, G. N., 1973: Zone of in¯uence±inner of Haidvogel, D., J. Wilkin, and R. Young, 1991: A semi-spectral prim- Georgia. J. Sediment. Petrol., 43, 207±214. itive equation ocean circulation model using vertical sigma and Blanton, J. O., 1996: Reinforcement of gravitational circulation by orthogonal curvilinear horizontal coordinates. J. Comput. Phys., wind. Buoyancy Effects on Coastal and Estuarine Dynamics. D. 94, 151±184. G. Aubrey and C. T. Friedrichs, Eds., Coastal and Estuarine Hubbard, M. E., 1999: Multidimensional slope limiters for MUSCL- Studies, 53, Amer. Geophys. Union, 47±58. type ®nite volume schemes on unstructured grids. J. Comput. Blumberg, A. F., 1994: A primer for Ecom-si. Tech. Rep., HydroQual, Phys., 155, 54±74. Inc., 66 pp. Kobayashi, M. H., J. M. C. Pereira, and J. C. F. Pereira, 1999: A ÐÐ, and G. L. Mellor, 1987: A description of a three-dimensional conservative ®nite-volume second-order-accurate projection coastal ocean circulation model. Three-Dimensional Coastal method on hybrid unstructured grids. J. Comput. Phys., 150, 40± Ocean Models, N. S. Heaps, Ed., Coastal and Estuarine Series, 45. Vol. 4, Amer. Geophys. Union, 1±16. Kraus, E. B., 1972: Atmosphere±Ocean Interaction. Clarendon Press, Casulli, V., and R. T. Cheng, 1991: A semi-implicit ®nite-difference 275 pp. model for three-dimensional tidal circulation. Proc. Second Int. Lynch, D. R., and C. E. Naimie, 1993: The M tide and its residual Conf. on Estuarine and Coastal Modeling, Tampa, FL, ASCE, 2 620±631. on the outer banks of the Gulf of Maine. J. Phys. Oceanogr., Chen, C., L. Z. Zheng, and J. O. Blanton, 1999: Physical processes 23, 2222±2253. controlling the formation, evolution, and perturbation of the low- Maday, Y., and A. T. Patera, 1989: Spectral element methods for the salinity front in the inner shelf off the southeastern United States: incompressible Navier±Stokes equations. State-of-the-Art Sur- A modeling study. J. Geophys. Res., 104, 1259±1288. veys on Computational Mechanics, A. K. Noor, Ed., ASME, 71± ÐÐ, J. Zhu, E. Ralph, S. A. Green, J. W. Budd and F. Y. Zhang, 143. 2001: Prognostic modeling studies of the Keweenaw Current in Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997a: Lake Superior. Part I: Formation and evolution. J. Phys. Ocean- A ®nite-volume, incompressible Navier±Stokes model for stud- ogr., 31, 379±395. ies of the ocean on parallel computers. J. Geophys. Res., 102, ÐÐ, R. C. Beardsley, P. J. S. Franks, and J. V. Keuren, 2002: In- 5753±5766. ¯uences of the diurnally varying heat ¯ux on strati®cation and ÐÐ, C. Hill, L. Perelman, and A. Adcroft, 1997b: Hydrostatic, quasi- residual circulation on Georges Bank. J. Geophys. Res., in press. hydrostatic and non-hydrostatic ocean modeling. J. Geophys. Cockburn, B., S. Hou, and C. W. Shu, 1990: TVB Runge±Kutta local Res., 102, 5733±5752. projection discontinuous Galerkin ®nite element method for con- Mellor, G. L., and T. Yamada, 1974: A hierarchy of turbulence closure servation laws. IV: The multidimensional case. Math Comput., models for planetary boundary layers. J. Atmos. Sci., 31, 1791± 54, 545±581. 1806. Dick, E., 1994: Introduction to ®nite-volume techniques in compu- ÐÐ, ÐÐ, and ÐÐ, 1982: Development of a turbulence closure tational ¯uid dynamics. Computational Fluid Dynamics, J. F. model for geophysical ¯uid problem. Rev. Geophys. Space Phys., Wendt, Ed., Springer-Verlag, 271±297. 20, 851±875. Dunstan, W. M., and L. P. Atkinson, 1976: Sources of new nitrogen ÐÐ, and A. E. Blumberg, 1985: Modeling vertical and horizontal

Unauthenticated | Downloaded 09/27/21 06:45 PM UTC 186 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 20

diffusivities with the sigma coordinate system. Mon. Wea. Rev., position, productivity and nutrient chemistry of a coastal ocean 113, 1379±1383. planktonic food web. Cont. Shelf Res., 13, 741±776. Naimie, C. E., 1996: Georges Bank residual circulation during weak Ward, M. C., 2000: An unsteady ®nite volume circulation model. and strong strati®cation periods: Prognostic numerical model Estuarine and Coastal Modeling: Proceedings of the 6th Inter- results. J. Geophys. Res., 101, (C3), 6469±6486. national Conference, November 3±5, 1999, New Orleans, Lou- Pomeroy, L. R., J. O. Blanton, G. A. Poffenhofer, K. L. V. Damm, isiana, M. L. Spaulding and H. L. Butler, Eds., ASCE, 17±33. P. G. Verity, H. L. Windom, and T. N. Lee, 1993: Inner shelf Wu, L., and D. B. Bogy, 2000: Use of an upwind ®nite volume method processes. Ocean Processes: U.S. Southeast Continental Shelf , to solve the air bearing problem of hard disk drives. Comput. D. W. Menzel, Ed., U.S. Dept. of Energy, 9±43. Mech., 26, 592±600. Reed, W. H., and T. R. Hill, 1973: Triangular and methods for the Zheng, L., and C. Chen, 2000: A 3D modeling study of the estaurine neutron transport equation. Tech. Rep. LA-UR-73-479, Los Al- system: An application to the Satilla River. Estuaries and Coastal amos Scienti®c Laboratory. Modeling: Proceedings of the 6th International Conference, No- Simpson, J. J., and T. D. Dickey, 1981a: Alternative parameterizations vember 3±5, 1999, New Orleans, Louisiana, M. L. Spaulding of downward irradiance and their dynamical signi®cance. J. and H. L. Butler, Eds., ACSE, 1128±1149. Phys. Oceanogr., 11, 876±882. ÐÐ, ÐÐ, M. Alber, C. Alexander, and J. O. Blanton, 2002a: A 3- ÐÐ, and ÐÐ, 1981b: The relationship between downward irradi- D modeling study of the Satilla River estuarine system. Part II: ance and upper ocean structure. J. Phys. Oceanogr., 11, 309± Suspended sedimentation. Estuaries, in press. 323. ÐÐ, ÐÐ, and J. Blanton, 2002b: A 3-D modeling study of the Verity, P. G., J. A. Yoder, S. S. Bishop, J. R. Nelson, D. B. Craven, Satilla River estuarine system. Part I: Simulation of ¯ooding/ J. O. Blanton, C. Y. Robertson, and C. R. Tronzo, 1993: Com- drying processes. Estuaries, in press.

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