Home » FVCOM » Model Validation » Case 08. ThreeDimensional WindDriven Flow in an Elongated, Rotating Basin
Case 08. ThreeDimensional WindDriven Flow in an Elongated, Rotating Basin
This test problem is motivated by the recent analytic solution for winddriven 3D homogeneous flow in a rotating elongated basin published by Clint Winant (SIO) (Winant, 2004). Kate Hedstrom (U. Alaska) first formulated the problem for the FORTRAN 77 version 1 of FVCOM and obtained mixed results, so Haosheng Huang (UMassD) volunteered to rerun the test problem using the then latest FVCOM FORTRAN 90 code (version 2.1.1) to determine if this updated version could accurately capture the analytic solution. The numerical solutions presented here show excellent agreement with Winant’s analytic solution, and demonstrate that the results are not sensitive to the choice of two types of unstructured grid. The model setup for this test problem are available at the FVCOM website (needs a password) for users who is interested in repeating this experiment. A brief description of this test problem and comparison between FVCOMbased numerical and analytical solutions is given below.  
Analytic Solution  
A linear theory for homogeneous winddriven circulation in a closed rectangular sloping basin was developed by Winant (2004). Consider a rectangular closed basin with length 2L and width 2B on an f plane shown in Fig. 1. The bathymetry of this basin is given by  
, (8.1) where X(x) is a function in the form of (8.2) and is a constant specified as 0.3% of the total length of the basin. The linearized governing equations for steady flow in this case can be simplified to (8.3) 

with boundary conditions specified as and , at (8.4) and , at (8.5) The analytical solutions to equations (8.3)(8.5) were derived by Winant (2004). The horizontal velocities are given by (8.6) and vertical velocity can be calculated using (8.7) where and ( Ekman number). The equations used to solve for and are described in detail in Winant (2004) (click here for PDF version of Winant’s paper on JPO). 

Design of numerical experiments  
The aim of this model experiment is twofold. One is to validate the FVCOM code versus the analytic solution. The other is to explore the sensitivity of the FVCOM code to different unstructured grid meshes used in the calculation. A key issue to remember is that the analytical solution was derived based on the noslip bottom boundary condition (see equation 8.5), while the bottom boundary condition in FVCOM (as in other ocean models) is given in terms of the bottom stress (with the default setup assuming that bottom stress is proportional to nearbottom velocity squared). In order to compare FVCOM results with the analytical solution, we have changed the bottom stress into a linear form proportional to the nearbottom velocity. This is still an approximate solution to the noslip analytical solution. The setup of FVCOM is constructed by 1) turning off all nonlinear, baroclinic pressure and horizontal diffusion terms in the momentum equations and 2) modifying the VDIF_UV.F and BROUGH.F to replace the bottom stress term using a linear form. The detailed modification list for this test can be seen in the setup note written by Haosheng Huang. 



In all experiments, K _{m} = 4.0 × 10 ^{3} m ^{2} s ^{1}. In FVCOM, the bottom stress is specified at the 0.5 sigma level above the bottom. An approximation of this stress can be simplified to be (8.8)  
Applying a linear bottom friction to equation (8.8), the linear bottom friction coefficient should be equal to K_{m}/∆h(x,y). In the numerical experiment using ROMS, Winant specified the bottom friction coefficient as K_{m}/∆h_{max} where ∆h_{max} is the maximum thickness of the vertical layer in the computational domain. Based on his estimation, K_{m}/∆h_{max}= 2.0 × 10^{3} m s^{1} . In order to make our numerical experiments conform closer to the noslip bottom condition used in the analytic solution, we have used K_{m}/∆h(x,y) to calculate the bottom friction coefficient. This is equivalent to assume that the velocity at the bottom is equal to zero. We have also run the model using the same setup used by Winant for his ROMS experiment. In summary, 4 experiments are conducted: Experiment 1: “Uniform” grid mesh and linear bottom friction coefficient specified using K_{m}/∆h(x,y); Experiment 2: “Star” grid mesh and linear bottom friction coefficient specified using K_{m}/∆h(x,y); Experiment 3: “Uniform’ grid mesh and linear bottom friction coefficient specified using K_{m}/∆h_{max}= 2.0 × 10^{3} m s^{1}; Experiment 4: “Star” grid mesh and linear bottom friction coefficient specified using K_{m}/∆h_{max}= 2.0 × 10^{3} m s^{1}. 

Model Results and Comparison with Analytic solution  
1.
Uniform grid Both experiments 1 and 3 show that FVCOMbased numerical solutions agree very well with the analytical solution in magnitudes and distributions of u, v, and w. An example of the numerical modelanalytical solution comparison is shown in Fig. 4, which illustrates the distribution of velocity on the crosssection at the middle of the basin. 

Fig. 4 The three velocity components (u, v, and w) along a midbasin cross section (x=0) in FVCOM experiments 1 and 3 and Winant’s analytic solution  
Differences of u, v, and w between modelpredicted and analytical solutions are very small. In a quantitative comparison, the difference in experiment 1 is an order of magnitude smaller than in experiment 3 (see Table 1 for detail). This indicates that the bottom friction coefficient specified as K_{m}/∆h(x,y) in experiment 1 presents a more accurate bottom boundary condition close to the noslip bottom condition specified in the analytical solution than the constant bottom friction condition using 2.0 × 10^{3} m s^{1} in experiment 3. One should expect a different shear profile of the velocity near the sloping bottom in this condition.  
Table 1 Maximum difference of the velocity components between the FVCOM results and Winant’s analytic solution along the midbasin cross section (x=0)  


∆U = MAX( U_{FVCOM} – U_{WINANT} ), ∆V = MAX( V_{FVCOM} – V_{WINANT} ), ∆W = MAX( W_{FVCOM} – W_{WINANT} )  
2.
Star grid The FVCOM results with “star” grids are almost identical to the results with “uniform” grids shown above (Fig. 5). The differences due to different parameterizations of the bottom friction coefficient are also evident in the star grid cases of experiments 2 and 4 (see Table 1). 

Fig. 5 The three velocity components (u, v, and w) along a midbasin cross section (x=0) in FVCOM experiments 2 and 4 and Winant’s analytic solution  
It should be pointed out here that the velocity calculated on “star” grids are distributed nonuniformly in the y direction. The graph plotting software tends to generate “zigzag” patterns along the sloping bottom due to linear interpolation. To make a more consistent comparison with analytical solution, we have output the velocity of the analytic solution at the same grid points used in the “star” grids. The images for both analytical and numerical solutions are produced using the same linear interpolation program for contour plots. One can easily see that the “zigzag” patterns are also shown in the analytic solution plots due to the image plotting software. Therefore, the difference of the current patterns shown in Figs. 4 and 5 is due to the graphic program for uniform and nonuniform grid interpolations.  
Summary  
FVCOM can reproduce accurately the dynamics and flow fields found in the Winant analytic solution for homogeneous winddriven steady circulation in a rotating elongated basin with sloping bottom. In this test problem, both the unstructured “uniform” and “star” grid meshes give quite similar FVCOM solutions 

Haosheng Huang, Changsheng Chen, Robert C. Beardsley, 07/07/2004 