Misc

# Case 08. Three-Dimensional Wind-Driven Flow in an Elongated, Rotating Basin

This test problem is motivated by the recent analytic solution for wind-driven 3-D 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 FVCOM-based numerical and analytical solutions is given below.
Analytic Solution
A linear theory for homogeneous wind-driven 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 no-slip 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 near-bottom 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 near-bottom velocity. This is still an approximate solution to the no-slip 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.
 Two kinds of unstructured grid mesh are used. One is called a “uniform” grid (Fig. 2) and the other is called a “star” gird (Fig. 3) In general, for all unstructured grids, “bigger” numerical errors might be expected when a control element is constructed with too many triangles, particularly at boundary cells. As part of the quality control step in building unstructured grids, it is always recommended to use no more than 8 triangles in a control element for either velocity or free surface computation. The “star “grid selected here includes 8 triangles in the tracer control element. Even for such a low-quality grid, the FVCOM solution in this experiment still matches well with the analytic solution. In both cases, the basin is 200 km long and 50 km wide (i.e., L = 100 km and B = 25 km), with x increment of 2 km and y increment 1 km. There are 25 equally spaced sigma layers in the vertical. The Coriolis parameter is taken as a constant value of f = 10 -4 s -1. The maximum water depth h 0 = 50 m. The wind stress is specified in the x-direction and wind forcing is linearly ramped up to a constant value of t x= 0.1 Pa over a 2-day period. FVCOM is run for a total of 50 days, the time increment is 20 seconds, with 20 external time steps in each internal time step. Calculation shows that the numerical solution approaches a steady state after about 10 days.  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 Km/∆h(x,y). In the numerical experiment using ROMS, Winant specified the bottom friction coefficient as Km/∆hmax where ∆hmax is the maximum thickness of the vertical layer in the computational domain. Based on his estimation, Km/∆hmax= 2.0 × 10-3 m s-1 . In order to make our numerical experiments conform closer to the no-slip bottom condition used in the analytic solution, we have used Km/∆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 Km/∆h(x,y);
Experiment 2: “Star” grid mesh and linear bottom friction coefficient specified using Km/∆h(x,y);
Experiment 3: “Uniform’ grid mesh and linear bottom friction coefficient specified using Km/∆hmax= 2.0 × 10-3 m s-1;
Experiment 4: “Star” grid mesh and linear bottom friction coefficient specified using Km/∆hmax= 2.0 × 10-3 m s-1.
Model Results and Comparison with Analytic solution
1.

Uniform grid

Both experiments 1 and 3 show that FVCOM-based numerical solutions agree very well with the analytical solution in magnitudes and distributions of u, v, and w. An example of the numerical model-analytical solution comparison is shown in Fig. 4, which illustrates the distribution of velocity on the cross-section at the middle of the basin. Fig. 4 The three velocity components (u, v, and w) along a mid-basin cross section (x=0) in FVCOM experiments 1 and 3 and Winant’s analytic solution
Differences of u, v, and w between model-predicted 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 Km/∆h(x,y) in experiment 1 presents a more accurate bottom boundary condition close to the no-slip 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 mid-basin cross section (x=0)
 ∆U cm s-1 ∆V cm s-1 ∆W 10-3 cm s-1 Experiment 1 0.48 0.35 0.88 Experiment 2 0.17 0.19 0.47 Experiment 3 4.2 1.8 5.4 Experiment 4 8.1 1.7 5.4
∆U = MAX( |UFVCOM – UWINANT| ), ∆V = MAX( |VFVCOM – VWINANT| ), ∆W = MAX( |WFVCOM – WWINANT| )
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 mid-basin 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 non-uniformly 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 non-uniform grid interpolations.
Summary
FVCOM can reproduce accurately the dynamics and flow fields found in the Winant analytic solution for homogeneous wind-driven 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