# Tide Module¶

There are two types of boundary conditions developed to incorporate tidal level and/or tidal current into model boundaries. One is an absorbing tidal boundary condition, which is essentially a sponge layer but takes into account a reference value, such as tidal level and tidal current, in the sponge regime. The other a so-called absorbing-generating boundary condition, which treats tidal level and/or tidal current as the background elevation/flow.

## Theory¶

The basic technique follows the sponge layer theory introduced by Larsen and Dancy (1983). Instead of attenuating the surface elevation and flow velocity to zero at the end of a sponge layer, we dampen shortwaves with respect to a reference level based on the method proposed by Chen et al. (1999). The dependent variables (\(\eta, u, v\)) are attenuated as

where \(( )_{ref}\) denotes the reference values which either tidal level or tidal current or both. \(C_s\) is the damping coefficient which is the same as that used in a sponge layer (see sponge layer section for detail), i.e.,

in which \(i\) is grid numbers, (\(i = 1, 2, ...\)).

## Absorbing tidal boundary condition¶

For the absorbing tidal boundary condition, the reference values (\(\eta_{ref}, u_{ref}, v_{ref}\)) are tidal levels and tidal current velocities specified either in input.txt or in a separate tide/surge file.

For a constant tidal condition, (\(\eta_{ref}, u_{ref}, v_{ref}\)) are constant, which are specified in input.txt. For example, in the case of tide_abs_1bc_constant in the directory of simple cases,

TIDAL_BC_ABS = T TideBcType = CONSTANT TideWest_ETA = 1.0 TideEast_ETA = 1.0

This is a 1D case with a tidal absorbing condition and constant tidal level specified above. The wavemaker can be any types of wavemaker available in the model. In this case, we used `WK\_REG`

,

WAVEMAKER = WK_REG DEP_WK = 8.0 Xc_WK = 150.0 Yc_WK = 0.0 Tperiod = 8.0 AMP_WK = 0.5 Theta_WK = 0.0 Delta_WK = 3.0

The figure shows a snapshot of surface elevation from the model. Note that the irregularity of wave surface is caused by the tidal propagation from both boundaries.

For a time-varying tidal condition, the tidal level and velocity are specified in a file named in `input.txt`

. For example, in the 2D case of `tide\_abs\_2bc\_data`

(same folder), tidal elevations are specified in two files, `tide\_data\_west.txt`

, `tide\_data\_east.txt`

, with the parameter `TideBcType`

set as DATA type,

TIDAL_BC_ABS = T TideBcType = DATA TideWestFileName = tide_data_west.txt TideEastFileName = tide_data_east.txt

The format of tidal data follows a list of ‘time, eta, u, v’ as shown in the figure below. In this case, a flat bottom of 8 m is applied in a 2D domain and a regular wavemaker is specified . The following figure demonstrates 2D and 1D section views of surface elevation at different times. Black solid lines denote tidal levels.

## Combined tidal and absorbing-generating boundary condition¶

The combined tidal and absorbing-generating boundary condition incorporates the solution of the linear wave theory and tidal elevation and velocity in the sponge layer.

The reference values (\(\eta_{ref}, u_{ref}, v_{ref}\)) are tidal levels and tidal current velocities specified either in input.txt or in a separate tide/surge file. Different from the tidal absorbing boundary condition, the reference values (\(\eta_{ref}, u_{ref}/v_{ref}\)) combine the tidal condition and wave solution, and specified over the entire computational domain. Inside the sponge layer, the differences between the reference values \(( )_{ref}\) and model solution \(( )_i\) are dampened by the sponge. Outside the sponge layer, independent variables are calculated directly from the model because \(C_s\) is 1.0. In this study, the west-side absorbing-generating boundary condition is implemented.

An example is provided In `/tide\_gen\_abs\_data/`

. Figure 4 shows the model setup with a west-side absorbing-generating boundary condition. In input.txt

WAVEMAKER = ABSORBING_GENERATING WAVE_DATA_TYPE = DATA WaveCompFile = wave_data.txt ... TIDAL_BC_GEN_ABS = T TideBcType = DATA TideWestFileName = tide_data_west.txt

The format of tidal data is the same as the tidal absorbing boundary condition. The model is set up in a 2D sloping beach domain. Figure 5 shows snapshots of surface elevation at different times.

## References¶

Chen, Q., Madsen, P.A., Basco, D.R., 1999. “Current Effects on Nonlinear Interactions of Shallow–Water Waves”. J. of Waterway, Port, Coastal, and Ocean Eng. 125 (4).

Larsen, J. and Dancy, H., 1983. “Open boundaries in short wave simulations – A new approach”. Coastal Eng. 7 (3), 285-297. DOI: 10.1016/0378-3839(83)90022-4.