JANUARY 2003 CHEN ET AL. 159 An Unstructured Grid, Finite-Volume, Three-Dimensional, Primitive Equations Ocean 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 Oceanography, 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 s-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 Sea, 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 oceans' 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- q 2003 American Meteorological Society Unauthenticated | Downloaded 09/27/21 06:45 PM UTC 160 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 20 cially in situations with sharp thermoclines or discon- ]u ]u ]u ]u 1 u 1 y 1 w 2 f y tinuous ¯ow. Unlike the differential form, the ®nite- ]t ]x ]y ]z volume method discretizes the integral form of the equa- tions, making it easier to comply with the conservation 1 ]P ]]u 52 1 K 1 F , (2.1) laws. Since these integral equations can be solved nu- r ]x ]z mu]z merically by the ¯ux calculation used in the ®nite-dif- o 12 ]y ]y ]y ]y ference method over an arbitrarily sized triangular mesh 1 u 1 y 1 w 1 fu (like those in a ®nite-element method), the ®nite-volume ]t ]x ]y ]z method seems to combine the best attributes of the ®- nite-difference method (for simple discrete computa- 1 ]P ]]y 52 1 K 1 F , (2.2) tional ef®ciency) and the ®nite-element method (for m y ro ]y ]z12]z geometric ¯exibility). ]P To our knowledge, a three-dimensional (3D), unstruc- 52rg, (2.3) tured grid, prognostic, primitive equation, ®nite-volume ]z ocean circulation model is not currently available in the ]u ]y ]w oceanographic community, although some efforts have 11 50, (2.4) ]x ]y ]z been made to develop a ®nite-volume formulation of the two-dimensional, barotropic shallow water equa- ]u ]u ]u ]u 1 u 1 y 1 w tions (Ward 2000). The MIT General Circulation model ]t ]x ]y ]z developed by Marshall et al. (1997a,b) is the ®rst 3D ®nite-volume ocean model. However, since this model ]]u 5 K 1 F , (2.5) currently relies on rectangular structure grids for hori- ]z12h ]z u zontal discretization, it is not suited to use for coastal ]s ]s ]s ]s ocean and estuarine domains with complicated geom- 1 u 1 y 1 w etries. Recently, we have developed a 3D unstructured ]t ]x ]y ]z grid, ®nite-volume coastal ocean model (called ]]s FVCOM). This new model has been applied to the Bohai 5 K 1 F , (2.6) Sea, a semienclosed coastal ocean, and the Satilla River, ]z12hs]z a Georgia estuary characterized by numerous tidal r 5 r(u, 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, y, and w are the x, y, z nite-volume model provides a better simulation of tidal velocity components; u is the potential temperature; s elevations and residual currents, especially around is- is the salinity; r is the density; P is the pressure; f is lands and tidal creeks. Both FVCOM and ECOM-si the Coriolis parameter; g is the gravitational accelera- show similar distributions of temperature and strati®ed tion; Km is the vertical eddy 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, Fy , Fu, 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 in section 6. Detailed expressions for the numerical ]q22]q ]q 2]q 2 1 u 1 y 1 w computation of individual terms in the momentum equa- ]t ]x ]y ]z tion are given in an appendix. ]]q2 5 2(Psb1 P 2«) 1 K q1 F q, (2.8) 2. The model formulation ]z12]z ]ql22]ql ]ql 2]ql 2 a. The primitive equations 1 u 1 y 1 w ]t ]x ]y ]z The governing equations consist of the following mo- Ä 2 mentum, continuity, temperature, salinity, and density W ]]ql 5 lE1 Psb1 P 2«1 K q1 F l, (2.9) equations: 1212E1 ]z ]z Unauthenticated | Downloaded 09/27/21 06:45 PM UTC JANUARY 2003 CHEN ET AL. 161 where q 2 5 (u9 2 1 y9 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 5 Km(uzz1 y ) and C 5 max , 0.0025 , (2.14) d 2 Pb 5 (gKhrz)/ro are the shear and buoyancy production zab terms of turbulent kinetic energy; «5q3/B l is the ln 1 12zo 2 turbulent kinetic energy dissipation rate; W 5 1 1 E 2l / ( L)2 is a wall proximity function, where L21 ( k 5 z 2 where k 5 0.4 is the von KaÂrmaÂn's constant and z is 21 (H z)21; 0.4 is the von KaÂrmaÂn constant; o z) 1 1 k 5 the bottom roughness parameter.