1. Introduction
Most of the world oceans' inner shelves and estuaries are characterized by a series of barrier island complexes, inlets, and extensive intertidal salt marshes. Such an irregular geometric ocean–estuarine system presents a challenge for oceanographers involved in model development even though the governing equations of oceanic circulation are well defined and numerically solvable in terms of discrete mathematics. Two numerical methods have been widely used in ocean models: 1) the finite-difference method (Blumberg and Mellor 1987; Haidvogel et al. 1991; Blumberg 1994) and 2) the finite-element method (Lynch and Naimie 1993; Naimie 1996). The finite-difference method is the simplest discrete scheme with an advantage of computational efficiency. Introducing an orthogonal or nonorthogonal curvilinear coordinate transformation into a finite-difference model can provide a moderate fitting of coastal boundaries, but these transformations are incapable of resolving the highly irregular estuarine geometries characteristic of numerous barrier island and tidal creek complexes (Blumberg 1994; Chen et al. 2001; Chen et al. 2002, manuscript submitted to J. Great Lakes Res.). The greatest advantage of the finite-element method is its geometric flexibility. Triangular meshes at an arbitrary size are used in this method and can provide an accurate fitting of the irregular coastal boundary. The P-type finite-element method (Maday and Patera 1989) or discontinuous Galerkin method (Reed and Hill 1973; Cockburn et al. 1990) has been introduced into the updated finite-element model to help improve computational accuracy and efficiency.
Recently, the finite-volume method has received considerable attention in the numerical computation of fluid dynamics (Dick 1994). The dynamics of oceanography comply with conservation laws. The governing equations of oceanic motion and water masses are expressed by the conservation of momentum, mass, and energy in a unit volume. When the equations are solved numerically, these laws cannot always be guaranteed, especially in situations with sharp thermoclines or discontinuous flow. Unlike the differential form, the finite-volume method discretizes the integral form of the equations, making it easier to comply with the conservation laws. Since these integral equations can be solved numerically by the flux calculation used in the finite-difference method over an arbitrarily sized triangular mesh (like those in a finite-element method), the finite-volume method seems to combine the best attributes of the finite-difference method (for simple discrete computational efficiency) and the finite-element method (for geometric flexibility).
To our knowledge, a three-dimensional (3D), unstructured grid, prognostic, primitive equation, finite-volume ocean circulation model is not currently available in the oceanographic community, although some efforts have been made to develop a finite-volume formulation of the two-dimensional, barotropic shallow water equations (Ward 2000). The MIT General Circulation model developed by Marshall et al. (1997a,b) is the first 3D finite-volume ocean model. However, since this model currently relies on rectangular structure grids for horizontal discretization, it is not suited to use for coastal ocean and estuarine domains with complicated geometries. Recently, we have developed a 3D unstructured grid, finite-volume coastal ocean model (called FVCOM). This new model has been 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 results obtained from a well-developed finite-difference model (called ECOM-si) and observational data, we find that the finite-volume model provides a better simulation of tidal elevations and residual currents, especially around islands and tidal creeks. Both FVCOM and ECOM-si show similar distributions of temperature and stratified tidal rectified and buoyancy-induced flows in the interior region in the Bohai Sea, but FVCOM seems to resolve the detailed thermal structure and flows around islands and complex coastal regions.
The remaining sections of this paper are organized as follows. The model formulation, design of unstructured grids, and discretization procedure are described in sections 2, 3, and 4, respectively. The model applications for the Bohai Sea and Satilla River are given and discussed in section 5, and a summary is provided in section 6. Detailed expressions for the numerical computation of individual terms in the momentum equation are given in an appendix.
2. The model formulation
a. The primitive equations
b. The governing equations in the σ coordinate
c. The 2D (vertically integrated) equations
The sea surface elevation included in the equations describes the fast-moving surface gravity waves. In the explicit numerical approach, the criterion for the time step is inversely proportional to the phase speed of these waves. Since the sea surface elevation is proportional to the gradient of water transport, it can be computed using vertically integrated equations. The 3D equations then can be solved under conditions with a given sea surface elevation. In this numerical method, called “mode splitting,” the currents are divided into external and internal modes that can be computed using two distinct time steps. This approach is used successfully in POM.
Recently, a semi-implicit scheme was introduced into POM, in which the sea surface elevation was computed implicitly using a preconditioned conjugate gradient method with no sacrifice in computational time (Casulli and Cheng 1991). This updated version of POM is called ECOM-si. The semi-implicit scheme cannot easily be applied to a finite-volume model since it is difficult to construct a linear positive symmetric algebraic matrix when unstructured triangular meshes are used. For this reason, we select the mode-splitting method to solve the momentum equations.
3. Design of the unstructured grids
To provide a more accurate estimation of the sea surface elevation, currents, and salt and temperature fluxes, the numerical computation is conducted in a specially designed triangular grid in which ζ, ω, s, θ, ρ, q2, q2l, H, D, Km, Kh, Am, and Ah are placed at nodes, and u, υ are placed at centroids. Variables at each node are determined by a net flux through the sections linked to centroids in the surrounding triangles with connection to that node. Variables at centroids are calculated based on a net flux through three sides of that triangle. The numerical code was written using Fortran 77 and can be run on a PC or workstation with Fortran 77 or above.
4. The discretization procedure
a. The 2D external mode
b. The 3D internal mode
The momentum equations are solved numerically using a simple combined explicit and implicit scheme in which the local change of the currents is integrated using the first-order accuracy upwind scheme. The advection terms are computed explicitly by a second-order accuracy Runge–Kutta time-stepping scheme is also incorporated in the updated version to increase the numerical integration to second-order accuracy. The procedure for this method is very similar to that described above for the 2D external mode. To provide a simple interpretation of the numerical approach for the 3D internal mode, we focus our description here only on the first-order accuracy upwind scheme. It should be noted here that the second-order accuracy Runge–Kutta time-stepping scheme is also incorporated in the model.
5. Model applications
To test our new unstructured grid, finite-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 multiple 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 characterized 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 semidiurnal (M2 and S2) and diurnal (K1 and O1) tides, which account for about 60% of the current variation and kinetic energy there. Since the tidally rectified residual flow is only substantial near the coast and islands in the Bohai Sea, geometric fitting is essential to providing a more accurate simulation of the tidal waves and residual flow. The Bohai Sea is connected to the Yellow Sea (on the south) through the Bohai Strait. Several islands located in the Strait complicate the water exchange between these two seas. Failing to resolve these islands leads to an underestimation of water transport through the strait. It also results in an unrealistic distribution of the tidal motion in the Bohai Sea due to alterations in the propagation paths of tidal waves. In addition, in the Bohai Sea, the tidally rectified residual flow is usually one order of magnitude smaller than the buoyancy- and wind-induced flows, except near the coast and around islands. In order to obtain a more accurate simulation of temperature and salinity, the model must be able to resolve the complex topography near the coast and around islands.
In the Satilla River, the M2 tidal current accounts for about 90% of the along-river current variation (Blanton 1996). Tidal advection and mixing also are the main physical processes controlling the spatial and temporal variations of biological and chemical materials in this estuary (Bigham 1973; Dunstan and Atkinson 1976; Pomeroy et al. 1993; Verity et al. 1993; Zheng and Chen 2000). Since the Satilla River estuary features numerous tidal creeks, failing to resolve these creeks would lead to under- or overestimating the tidally rectified flow. This in turn would cause water transport in the river to be miscalculated. This can be seen clearly in the comparison between the finite-difference and finite-volume model results of the Satilla River given below.
a. The Bohai Sea
The finite-difference model used in this comparison is ECOM-si, which is an updated version of POM. The model domains for FVCOM and ECOM-si are shown in Fig. 4, both of which have their open boundaries in the Yellow Sea about 150 km south of the Bohai Strait. In FVCOM, the horizontal resolution is about 2.6 km around the coast and about 15–20 km in the interior and near the open boundary. In ECOM-si, a uniform horizontal 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 σ 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 semidiurnal (M2 and S2) and diurnal (O1 and K1) tidal elevations and phases at the open boundary. The sea level 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 stratification. The initial temperature was specified as a vertical linear function with 25°C at the surface and 15°C at a depth of 75 m. The salinity was specified as a constant value of 30 psu. The time step was 186.3 s, which corresponded to 240 time steps over the M2 tidal cycle.
The model-predicted time series of surface elevation and currents at each grid point was fitted by a least squares harmonic analysis method. The resulting coamplitude and cophase of each tidal constituent are shown in Figs. 5 and 6. The model–data comparisons of tidal amplitudes and phases at tidal measurement stations are given in Tables 1–4. Although both FVCOM and ECOM-si show that M2 and S2 tidal waves propagate counterclockwise around the coast like a Kelvin wave, the distributions of tidal amplitudes and phases predicted by these two models differ significantly. FVCOM predicts two nodes of the M2 and S2 tides in the Bohai Sea: one is near the mouth of the Yellow River on the southwestern coast, and the other is located offshore of Qinhuangdao on the northwestern coast. These two nodes, however, shift onshore in the case of the ECOM-si, especially for the M2 tide. The FVCOM-predicted maximum amplitudes of the M2 and S2 tides are about 130 cm in Liaodong Bay and 100 cm in Bohai Bay, both of which are about 10–20 cm higher than those predicted by the ECOM-si. Both FVCOM and ECOM-si show similar structures for the K1 and O1 tides, but the model-predicted amplitude of the K1 tide is higher in the case with FVCOM than in the case with ECOM-si.
The comparison between observed and model-predicted amplitudes and phases of semidiurnal tides at tidal measurement stations around the Bohai Sea shows a better agreement in the case with FVCOM than in the case with ECOM-si, especially for the M2 tidal constituent in Bohai Bay and Liaodong Bay. The standard deviation for the M2 and S2 tidal simulations is 6.0 and 5.8 cm in amplitude and 18.9° and 29.9° in phase, respectively, in the case with FVCOM. However, they are 16.6 and 5.9 cm in amplitude and 41.2° and 42.9° in phase in the case with ECOM-si (Tables 1 and 2). The FVCOM is more capable of predicting the amplitude and phase of semidiurnal tides in Liaodong Bay and Bohai Bay than ECOM-si. However, no significant differences are found for the K1 and O1 tides in both the FVCOM and ECOM-si models (Tables 3 and 4).
Both FVCOM and ECOM-si predict relatively weak tidally rectified residual currents in the Bohai Sea except near the coast and around islands. In the Bohai Strait, for example, the FVCOM model shows multiple around-island residual flow patterns. These patterns are not well predicted by the ECOM-si model because of poor resolution around the islands (Fig. 7). Although both FVCOM and ECOM-si models show an eastward residual flow along the southern coast in the Bohai Sea, the current is trapped near the coast and is much stronger in the case with FVCOM than in the case with ECOM-si. Similar disparities also are found around the islands in the eastern coast and Bohai Bay.
For the same initial distribution of temperature, the distributions of the temperature predicted by FVCOM and ECOM-si on the 10th model day are similar in the interior but differ significantly around the coast and islands. In the horizontal, both models predict a tidal mixing front around the 15-m isobath and a relatively uniform temperature in the interior (Fig. 8). In the vertical, they also show the similar tidal mixing height above the bottom on sections 1 and 2 (Fig. 9). The major difference is that the cross-frontal gradient of temperature around the 15-m isobath is relatively larger in the case with FVCOM than in the case with ECOM-si. Also, the model-predicted depth of the thermocline in section 1 is shallower in the case with FVCOM than in the case with ECOM-si.
Disparity in the field of the temperature between FVCOM and ECOM-si is believed due to the difference of the accuracy of geometric matching between these two models. At site I (in the interior), for example, both FVCOM and ECOM-si show that the temperature at the surface and middle depth remains almost unchanged during the first 10 model days, while the temperature near the bottom starts mixing up after 1 day (the model boundary forcing is ramped up from zero to full amplitude over the first 24 h of model integration) and reaches to an equilibrium state after 4 model days (Fig. 10a). FVCOM shows relatively stronger mixing on the second model day, which probably is caused by the difference in horizontal resolution and water depth interpolated from irregularly distributed dataset between these two models. At site 2 (around an island close to the Bohai Strait), the near-surface temperature decreases slightly with time in the case with ECOM-si, but drops more rapidly with time and also oscillates periodically after the fourth model day in the case with FVCOM (Fig. 10b). Although temperature at middle depth and near the bottom predicted by these two models tends to mix up after 10 model days, the mixing rate seems faster in FVCOM than in ECOM-si (Fig. 10b).
This is not a surprising result since the topography around the island is resolved well in FVCOM but not in ECOM-si. If we believe that both FVCOM and ECOM-si have the same numerical accuracy, then we could conclude here that poor matching of the complex coastal geometries in the finite-difference model would underestimate mixing around the coast, which would eventually lead to the unrealistic distribution of the temperature in the interior, especially in a semienclosed coastal ocean like the Bohai Sea. Also, we learn from site 2 that the mismatch in the island geometry would filter a relatively large tidal oscillation near the surface, which tends to produce significant mixing near the surface under conditions with no heat flux.
b. The Satilla River
The model grids of FVCOM and ECOM-si for the Satilla River are shown in Fig. 11. The horizontal resolution of ECOM-si is 100 m in the main channel of the river and up to 2500 m near the open boundary in the inner shelf. Similar sizes of unstructured grids are used in FVCOM. In both models, the vertical is divided into 10 uniform σ layers, which correspond to a vertical resolution of less than 0.5 m in most areas inside the river. The models were driven by the same semidiurnal M2 tidal forcing at the open boundary. The harmonic constants of the M2 tidal forcing were specified using the tidal elevations and phases predicted by the inner shelf South Atlantic Bight (SAB) tidal model [developed and calibrated by Chen et al. (1999)]. No stratification or river discharge is included in this model comparison experiment.
The model results show a significant difference in the along-river distribution of the M2 tidal amplitude between FVCOM and ECOM-si (Fig. 12). The observed amplitude of the M2 tidal constituent is 94.7 ± 1.3 cm at site 1, gradually increases to 99.4 ± 1.4 cm at site 4, and then decreases to 96.0 ± 1.3 cm at site 5. At sites 6 and 7 in the southern and northern branches separated at the upstream end of the main river channel, the observed amplitudes are 92.2 ± 1.3 cm and 96.4 ± 1.3 cm, respectively. The amplitude of the sea level predicted by ECOM-si increases upstream, with values significantly higher than the observed values at sites 4 and 5. Since ECOM-si fails to resolve the two river branches at the upstream end of the main channel, water flooding up the river tends to accumulate there. In contrast, FVCOM not only predicts the same trend of the M2 tidal amplitude as the observations from site 4 to 7, but also their values agree with each other within measurement uncertainty. In addition, FVCOM shows higher values of the amplitude than the observations at sites 1–3, which is believed due to the flooding/drying process over the intertidal zone around the mouth of the river.
Zheng et al. (2002b) incorporated a 3D wet/dry point treatment method into ECOM-si and used it to simulate the amplitude and phase of the M2 tidal constituent in the Satilla River. They found that the flooding/drying process plays a key role in simulating tidal elevation and currents in the main river channel. Including the intertidal zone in the ECOM-si did show a significant improvement in the simulation of tidal elevation at site 5, but it still fails to provide reasonable values of the amplitude at sites 6 and 7. To make the model-predicted tidal elevation match the observed value at measurement sites, Zheng et al. (2002b) tuned the model by adjusting the bottom roughness zo. Since we have not yet added flooding/drying to FVCOM, the model comparison made is between both models without this process.
Tidal currents computed by FVCOM and ECOM-si also differ significantly, especially around the estuary–tidal creek area (Figs. 13 and 14). FVCOM shows a relatively strong tidal current near both the southern and northern coasts, with a substantial inflow to and outflow from tidal creeks during flood and ebb tides, respectively. These patterns are not resolved in ECOM-si. FVCOM predicts a stronger along-coast residual flow near the tidal creek, which intensifies the topographically induced eddylike residual circulation cell on the eastern side of the tidal creek (Fig. 15a). Although this eddylike residual circulation cell is also predicted in ECOM-si, it is much weaker and the velocity is symmetrically distributed relative to its center (Fig. 15b).
6. Discussion and summary
An unstructured grid, finite-volume, three-dimensional primitive equation coastal ocean model (FVCOM) has been developed for the study of coastal and estuarine circulation. This model combines the advantages of the finite-element method for geometric flexibility and finite-difference method for simple discrete computational efficiency. The numerical experiments in the Bohai Sea and Satilla River demonstrate that this model provides a more accurate simulation of tidal currents and residual flow in coastal ocean and estuarine settings where multiple islands, inlets, and tidal creeks exist. Because of a better fitting of the geometric complex in FVCOM, this model should provide a more accurate representation of water mass property variability and the advection and mixing of passive tracers around the coast.
FVCOM and ECOM-si show similar accuracy in the tracer simulation experiments except around complex topographies. Regarding the finite-difference approach, the most significant improvement provided by FVCOM is the geometric flexibility with unstructured grids. Recently, some model experiments were conducted with FVCOM, ECOM-si, and POM for two idealized cases with analytic solutions: free long gravity waves in a circular lake, and tidal wave resonance in a simple semienclosed channel. The results show in the first case that poor resolution of the curved coastal geometry causes both unwanted wave damping and a time-dependent phase shift. In the second case, the near-resonance behavior is strongly influenced by channel shape irregularies. This may explain why FVCOM provides a more accurate simulation for the amplitude of the M2 tidal constituent in Bohai Bay. A manuscript describing these and other idealized model comparisons is in preparation.
The goal of this paper is to introduce the unstructured grid, finite-volume numerical approach to the coastal ocean community. We fully understand that more experiments and comparisons with analytical solutions and other models must be made in order to validate the usefulness and reliability of this new finite-volume ocean model for the study of coastal and estuarine circulation and ecosystem dynamics.
Recently, a wet/dry point treatment technique was introduced into FVCOM. It is now being tested in the Satilla River, an estuary characterized by intensive intertidal salt marshes. Also, a Lagrangian particle tracking 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 developed. 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 encouragement and support of this model development effort. Without the continuous financial support of the Georgia Sea Grant College Program, this model development would not have been possible. We also want to thank A. Blumberg, who gave us permission to use ECOM-si during the last 10 years. Our experience with ECOM-si has helped us build a solid foundation in modeling, which in turn has directly benefited us in developing FVCOM. G. Davidson (Georgia Sea Grant College Program) provided editorial help on this manuscript. His assistance is greatly appreciated. Three anonymous reviewers provided many critical comments and constructive suggestions, which really helped us to improve the model–data comparison and clarify the final manuscript.
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APPENDIX
The Discrete Form of the 2D External and 3D Internal Modes
The 2D external mode
The 3D internal mode
The unstructured grid for the finite-volume model
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
The unstructured grid for the finite-volume model
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
The unstructured grid for the finite-volume model
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;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 filled triangles are the sections and site used for model comparisons
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;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 filled triangles are the sections and site used for model comparisons
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;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 filled triangles are the sections and site used for model comparisons
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
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 Oceanography, Savannah, GA.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
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 Oceanography, Savannah, GA.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
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 Oceanography, Savannah, GA.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Unstructured and curvilinear grids of the Bohai Sea for FVCOM and ECOM-si.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Unstructured and curvilinear grids of the Bohai Sea for FVCOM and ECOM-si.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Unstructured and curvilinear grids of the Bohai Sea for FVCOM and ECOM-si.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
The charts of model-predicted coamplitudes and cophases of the M2 and S2 tides in the Bohai Sea. (right) ECOM-si and (left) FVCOM.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
The charts of model-predicted coamplitudes and cophases of the M2 and S2 tides in the Bohai Sea. (right) ECOM-si and (left) FVCOM.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
The charts of model-predicted coamplitudes and cophases of the M2 and S2 tides in the Bohai Sea. (right) ECOM-si and (left) FVCOM.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
The charts of model-predicted coamplitudes and cophases of the K1 and O1 tides in the Bohai Sea. (right) ECOM-si and (left) FVCOM
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
The charts of model-predicted coamplitudes and cophases of the K1 and O1 tides in the Bohai Sea. (right) ECOM-si and (left) FVCOM
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
The charts of model-predicted coamplitudes and cophases of the K1 and O1 tides in the Bohai Sea. (right) ECOM-si and (left) FVCOM
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of (a) FVCOM and (b) ECOM-si predicted surface residual current vectors around the islands close to the Bohai Sea Strait.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of (a) FVCOM and (b) ECOM-si predicted surface residual current vectors around the islands close to the Bohai Sea Strait.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of (a) FVCOM and (b) ECOM-si predicted surface residual current vectors around the islands close to the Bohai Sea Strait.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of model-predicted near-surface (1/2 σ-level below the surface) temperature averaged over a M2 tidal cycle of the 10th model day. (top) FVCOM and (bottom) ECOM-si.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of model-predicted near-surface (1/2 σ-level below the surface) temperature averaged over a M2 tidal cycle of the 10th model day. (top) FVCOM and (bottom) ECOM-si.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of model-predicted near-surface (1/2 σ-level below the surface) temperature averaged over a M2 tidal cycle of the 10th model day. (top) FVCOM and (bottom) ECOM-si.
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Vertical distributions of temperature averaged over a M2 tidal cycle of the 10th model day on cross-sea sections 1 and 2. (right) ECOM-si and (left) FVCOM
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Vertical distributions of temperature averaged over a M2 tidal cycle of the 10th model day on cross-sea sections 1 and 2. (right) ECOM-si and (left) FVCOM
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Vertical distributions of temperature averaged over a M2 tidal cycle of the 10th model day on cross-sea sections 1 and 2. (right) ECOM-si and (left) FVCOM
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Time series of model-predicted temperature at the surface (1/2 σ level below the surface), middle-depth, and bottom (1/2 σ level above the bottom) at selected sites I and II (shown in Fig. 2) in the Bohai Sea. Solid line: FVCOM; dashed line: ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Time series of model-predicted temperature at the surface (1/2 σ level below the surface), middle-depth, and bottom (1/2 σ level above the bottom) at selected sites I and II (shown in Fig. 2) in the Bohai Sea. Solid line: FVCOM; dashed line: ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Time series of model-predicted temperature at the surface (1/2 σ level below the surface), middle-depth, and bottom (1/2 σ level above the bottom) at selected sites I and II (shown in Fig. 2) in the Bohai Sea. Solid line: FVCOM; dashed line: ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Unstructured and curvilinear grids of the Satilla River for FVCOM and ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Unstructured and curvilinear grids of the Satilla River for FVCOM and ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Unstructured and curvilinear grids of the Satilla River for FVCOM and ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Comparison between model-predicted (FVCOM and ECOM-si) and observed amplitudes of M2 tidal elevation at seven measurement sites shown in Fig. 3
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Comparison between model-predicted (FVCOM and ECOM-si) and observed amplitudes of M2 tidal elevation at seven measurement sites shown in Fig. 3
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Comparison between model-predicted (FVCOM and ECOM-si) and observed amplitudes of M2 tidal elevation at seven measurement sites shown in Fig. 3
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of the near-surface M2 tidal currents at the maximum flood tide. (top) FVCOM and (bottom) ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of the near-surface M2 tidal currents at the maximum flood tide. (top) FVCOM and (bottom) ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of the near-surface M2 tidal currents at the maximum flood tide. (top) FVCOM and (bottom) ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of the near-surface M2 tidal currents at the maximum ebb tide. (top) FVCOM and (bottom) ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of the near-surface M2 tidal currents at the maximum ebb tide. (top) FVCOM and (bottom) ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distributions of the near-surface M2 tidal currents at the maximum ebb tide. (top) FVCOM and (bottom) ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distribution of model-predicted surface residual current vectors in the selected area of the Satilla River. (top) FVCOM and (bottom) ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distribution of model-predicted surface residual current vectors in the selected area of the Satilla River. (top) FVCOM and (bottom) ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Distribution of model-predicted surface residual current vectors in the selected area of the Satilla River. (top) FVCOM and (bottom) ECOM-si
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Fig. A1. The illustration of the local coordinates
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Fig. A1. The illustration of the local coordinates
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Fig. A1. The illustration of the local coordinates
Citation: Journal of Atmospheric and Oceanic Technology 20, 1; 10.1175/1520-0426(2003)020<0159:AUGFVT>2.0.CO;2
Model–data comparison of the M2 tidal amplitude and phase in the Bohai Sea
Model–data comparison of the S2 tidal amplitude and phase in the Bohai Sea
Model–data comparison of the K1 tidal amplitude and phase in the Bohai Sea
Model–data comparison of the O1 tidal amplitude and phase in the Bohai Sea
