# 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

(112)$\eta_i = \eta_{ref} + (\eta_i - \eta_{ref})/C_s$
(113)$u_i = u_{ref} + (\eta_i - u_{ref})/C_s$

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.,

(114)$C_s = \alpha_s^{\gamma_s^{(i-1)}}$

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.

Fig. 1 Output from the case of tidal absorbing boundary condition.

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.

Fig. 2 Layout of tidal absorbing boundary (west and east).

Fig. 3 Case: /simple\_cases/tide\_abs\_2bc\_data/. Demonstration of 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.

Fig. 4 Layout of generating and absorbing boundary (left only)

Fig. 5 Case: /simple_cases/tide_gen_abs_data/. Thick dashed lines represent tidal levels. Thin black line denotes the beach slope.