Misc

Case 03. Tidal resonance in a semi-enclosed rectangular channel

1. Analytical Solution
The geometric structure of the channel is shown in Fig. 1, where is the water depth that decreases linearly toward the end of the channel, is the water depth of the open boundary of the channel, and are the distances from the origin to the end and mouth of the channel, and B is the width of the channel. Fig 1. The illustration of the semi-enclosed channel for the tidal wave test case. O. B.: Open boundary; B. W.: Boundary wall at the end of the channel; B is the width of the channel; and Ho is the mean depth at the open boundary (3.1)
Rewriting that and (3.1) yields, (3.2)
Specifying a periodic tidal forcing with amplitude of A at the mouth of the channel, i.e., (3.3)
and a no-flux boundary condition at the end of the channel, i.e., (3.4)
the analytical solution of (3.2) is given as (3.5)
where (3.6)
J0 and Y0 are zero- and first-order Bessel functions, and .
2. Design of the Numerical Experiment
The nature of the oscillation described in (3.5) depends on the geometry of the semi-enclosed channel. Two cases are tested here: 1. Normal (non-resonance) case: B = 5km, H0=20 m, L = 290 km, and H(L1) = 10 m; 2. Near-resonance case: B = 5km, H0=20 m, L = 290 km, and H(L1) = 0.67 m Both cases are driven by the M2 tidal forcing [frequency: ω = 2Π / (12.42 X 3600 sec); amplitude: = 1 cm] at the open boundary Fig 2. Unstructured triangular and structured rectangular grids used for FVCOM and POM/ECOM-si. The horizontal resolution is 2.5 km.
Numerical experiments are designed for a 2-D case, in which no advections, cross-channel variation, and mixing are taken into account. Numerical grids of FVCOM and POM/ECOM-si are constructed by triangular and square meshes with a horizontal resolution of 2.5 km, respectively. Since the cross-channel current is zero everywhere, only two triangular and square grid cells in the cross-channel section are needed to calculate the water elevation and along-channel transport for either FVCOM or POM/ECOM-si. To avoid artificial oscillations caused by an impulse of tidal forcing at open boundary, the elevation and current at each grid at the initial are specified according to the analytical solution
3. Results
 In the normal case, given small tidal amplitude of 1 cm at the mouth of the channel, is characterized with a node point at the middle of the channel and a maximum value of 1.2 cm at the end of the channel. No resonance can happen in this case. The amplitude and phase of the M2 tidal wave computed by FVCOM, POM, and ECOM-si are identical, all of them are in accurate agreement with the analytical solution (Fig. 3). The analytical solution describes a standing wave with a node point at the center of the channel. This feature is accurately reproduced by all the three models no matter what numerical schemes are used. Fig 3. Comparison of the model-predicted and analytical amplitudes and phases in the along-channel direction under a non-resonance geometric condition for FVCOM, POM and ECOM-si. The solid line is the analytical solution and dashed line is the model simulation. The origin of the coordinate is located at the end of the channel. In this case, the initial distributions of currents and sea surface are specified using the analytical solution.
In the near-resonance case, however, the elevation computed by POM grows exponentially with time (Fig. 4). Numerical instability eventually causes the model to blow up after 30 tidal cycles, even though the basic pattern of the along-channel distribution of the elevation remained unchanged. This instability can not be suppressed by reducing the time step. The semi-implicit implicit scheme used in ECOM-si ensures numerical stability, this method, however, causes a considerable numerical oscillation relative to the exact solution and reverses the phase after 20 tidal cycles. Unlike POM and ECOM-si, FVCOM remains numerically stable at all times. The amplitude of the elevation computed by FVCOM accurately matches the exact solution, although the phase at the node point shows a small oscillation relative to the exact value.
 Theoretically speaking, for these particular rectangular channel cases, the finite-volume and finite-difference approaches should be identical, since they all are designed to conserve the mass in individual volumes. The difference in the performances of POM, ECOM-si, and FVCOM for the near-resonance case is believed to be due to particular numerical methods used in these models. The standard version of POM uses a central difference scheme for both time integration and spatial gradients of surface elevation and volume transport. Although it remains a second order accuracy for numerical computation, it requires time and space smoothing to avoid numerical instability. Fig 4. Comparison of the model-predicted and analytical amplitudes and phases in the along-channel direction under a near-resonance geometric condition for FVCOM, POM and ECOM-si. The solid line is the analytical solution and dashed lines are the model results. TC: Tidal cycle. In this case, the initial distributions of currents and sea surface are specified using the analytical solution.
Since this model depends heavily on an artificial smoothing procedure to suppress the growth of unbalanced masses, it is always questioned whether or not this approach is suitable for a long-term integration regarding an issue of mass conservation. ECOM-si uses a semi-implicit scheme to release the restriction of time step used to calculate the surface gravity waves. This method, however, could lead to a numerical decay of tidal energy for improper selection of time step. It is clear that the numerical method used in ECOM-si fails to simulate the tidal wave under near-resonance condition in the shallow water, even though it could always keep numerical computations stable. FVCOM is solved numerically using an integrated form of the momentum equations with an approach of flux estimation at second-order accuracy (Chen et al., 2003). This method seems better than methods used for POM and ECOM-si for an unusual case with large tidal oscillations in shallow water.