Tsunami Damping due to Bottom Friction Considering Flow Regime Transition and Depth-Limitation in a Boundary Layer

Abstract

According to recent investigations on bottom boundary layer development under tsunami, a wave boundary can be observed even at the water depth of 10 m, rather than a steady flow type boundary layer. Moreover, it has been surprisingly reported that the tsunami boundary layer remains laminar in the deep-sea area. For this reason, the bottom boundary layer under tsunami experiences two transitional processes during the wave shoaling: (1) flow regime transition in a wave-motion boundary layer from laminar to the turbulent regime, and (2) transition from non-depth-limited (wave boundary layer) to depth-limited boundary layer (steady flow boundary layer). In the present study, the influence of these two transition processes on tsunami wave height damping has been investigated using a wave energy flux model. Moreover, a difference of calculation results by using the conventional steady flow friction coefficient was clarified.

1. Introduction

Since the bottom boundary layer under a wave motion is greatly connected with important coastal phenomena such as sediment movement and wave energy attenuation, there have been numerous investigations on the wave boundary layer in terms of experimental, theoretical, and numerical approaches. The wave boundary layer is characterized by an extremely thin thickness with a steep velocity gradient in the immediate vicinity of the sea bottom, unlike a steady flow in an open channel. For this reason, a U-shaped oscillating tunnel has widely been utilized in laboratory experiments, which enables generation of an oscillating period similar to real scale wave motion. It was Jonsson [1,2] who first applied this experimental facility to investigate the turbulent structure of sinusoidal wave boundary layers. Thereafter, many researchers utilized this generation system and obtained highly useful results under sinusoidal oscillatory flows, such as Sleath [3] and Jensen et al. [4]. In more recent years, an oscillating tunnel has widely been utilized to investigate more complicated oscillatory turbulent flows, including acceleration-skewed oscillatory flow [5], wave-current coexistent motion [6], and irregular wave motion [7].

Meanwhile, for investigating turbulent structure under long period waves in laboratory experiments, it is common to use a shallow oscillatory open-channel flow, such as Yalin and Russell [8], Knight and Ridgway [9], Chen et al. [10] and Larsen et al. [11], except Tanaka et al. [12] who used an oscillating water tunnel. Another approach dealing with long wave boundary layer has been a numerical one such as Cereki and Rodi [13], Li et al. [14], Williams and Fuhrman [15], Kaptein et al. [16], and Tanaka et al. [17]. In particular, Williams and Fuhrman [15] reported that the behavior of the tsunami boundary layer is highly similar to that of wind-generated waves, as the tsunami boundary layer thickness $\delta$ is extremely thinner than the water depth $h$. More recently, Tinh and Tanaka [18] and Tanaka and Tinh [19] obtained a similar result through an investigation for a shoaling tsunami from a tsunami source to a shallow sea area. Furthermore, Tanaka et al. [20] pointed out the necessity to recognize two transitions during the tsunami shoaling process, as shown in Figure 1a: (1) transition of flow regime in a wave boundary layer (Figure 1b) [21,22], and (2) transition from wave friction zone (WFZ) to steady friction zone (SFZ) [20,23]. In this study, the wave boundary layer thickness is defined based on Jensen et al. [4] as illustrated in Figure 1b.

However, the influence of such two types of transitional behavior on resultant tsunami damping has never been investigated. In this study, investigation of tsunami height attenuation considering these two transition processes shown in Figure 1a is carried out using a wave flux method combined with an equation for wave friction coefficient, which spans all flow regimes [24]. In addition, the conventional steady friction factor and wave friction coefficient for rough-turbulent regime [25] are applied to investigate the importance of recognizing transition behavior.

2. Methodology

2.1. Wave Flux Method

In order to include the viscous effect in Boussinesq equations for long waves, Liu and Orfila [26] proposed equations expressed in terms of differential-integral equations. Liu [27] and Orfila et al. [28] extended this method to the turbulent regime using an eddy viscosity model. Tinh and Tanaka [29] developed a numerical method for tsunami simulation that elaborates a wave friction factor by implementing a correction method into the conventional 1D shallow water equation. In this study, an energy flux method in collaboration with a wave friction coefficient is employed instead of momentum equations.

At first, ignoring energy loss due to bottom friction, wave energy conservation is described by the following equation.

$$\frac{dW}{dx}=0$$

(1)

where $W$: the wave energy flux, and $x$: the horizontal coordinate taken positive in the direction of wave propagation. According to the small amplitude long wave theory, $W$ is expressed as follows.

$$W=E{c}_g=\frac{1}{8}\rho g{H}^2\sqrt{gh}$$

(2)

where $E$: the wave energy, $c_g$: the group velocity, $\rho$: the sea water density, $g$: the gravitational acceleration, $H$: the wave height, and $h$: the water depth. Substituting Equation (2) into Equation (1), we obtain the following ordinary differential equation.

$$\frac{dH}{dx}=-\frac{H}{4h}\frac{dh}{dx}$$

(3)

By integrating Equation (3) with respect to x, the well-known Green’s formula can be derived.

$$\frac{H}{H_0}={\left(\frac{h_0}{h}\right)}^{1/4}$$

(4)

where h₀ and H₀ are the depth and wave height at tsunami source point, respectively.

Next, including the energy loss due to bottom shear stress averaged over a wave cycle, $D_f$, the following equation may be used instead of Equation (3) [30,31].

$$\frac{dH}{dx}=-\frac{H}{4h}\frac{dh}{dx}-D_f$$

(5)

where the energy loss by bottom friction is expressed by the following formula [30,31].

$$D_f=\frac{2}{3\pi }\rho f{U}_m^3$$

(6)

where $f$: the friction coefficient and $U_m$: the maximum flow velocity induced by the wave motion. Here, the relationship between the bottom shear stress ${\tau }_0$ and the friction coefficient is given by the following equation.

$${\tau }_0\left(t\right)=\frac{\rho }{2}fU\left(t\right)\left|U\left(t\right)\right|$$

(7)

where $U\left(t\right)$: the instantaneous velocity induced by wave motion. In Equation (7), the phase difference between a bottom shear stress and the flow velocity is ignored, as it is negligibly small under wave motion, especially under a turbulent regime [4]. Moreover, the maximum flow velocity in Equation (6) is obtained from the linear long wave theory.

$$U_m=\frac{H}{2}\sqrt{\frac{g}{h}}$$

(8)

Substituting Equations (2), (6), and (8) into Equation (5), we obtain the following governing equations considering both wave shoaling and energy loss due to bottom shear stress.

$$\frac{dH}{dx}=-\frac{H}{4h}\frac{dh}{dx}-\frac{f{H}_{}^2}{3\pi h^2}$$

(9)

For laminar wave boundary layer, an analytical solution for wave damping under sinusoidal wave can be obtained [32]. For turbulent conditions, meanwhile, typical values such as $f=0.02$ and $f=0.002$ have been employed [33].

To deal with the complicated transitional processes consisting of (1) transition of flow regime, and (2) transition from WFZ to SFZ, a numerical analysis is conducted using Equation (9).

2.2. Friction Factor

The friction coefficient f in Equation (9) is obtained from seven different methods, as listed in

Table 1. Friction coefficients used to integrate Equation (9).

Method	Used Friction Factor
1	$$f=0 \left(\mathrm{Green}’\mathrm{s}\mathrm{law}\right)$$ (10)
2	$$f=0.002$$ (11)
3	$$f=0.02$$ (12)
4	$$f=\frac{2{\kappa }^2}{{\left\{ln\left(\frac{h}{z_0}\right)-1\right\}}^2}$$ (13)
5	$$f=\frac{2g{n}^2}{h^{1/3}} (n=0.025)$$ (14)
6	If δ/h < 1 (non-depth-limited), f = fw for rough-turbulent regime by Tanaka [25] $$f_w=\mathit{exp}\left\{-7.53+8.07{\left(\frac{30a_m}{k_s}\right)}^{-0.100}\right\}$$ (15) If δ/h ≥ 1 (depth-limited), Equation (13).
7	If δ/h < 1 (non-depth-limited), f = fw from the full-range equation by Tanaka and Thu [24] $$f_w=f_2\left\{f_1{f}_{w\left(L\right)}+\left(1-f_1\right)f_{w\left(S\right)}\right\}+\left(1-f_2\right)f_{w\left(R\right)}$$ (16) where $$f_{w\left(L\right)}=\frac{2}{\sqrt{R_e}}$$ (17) $$f_{w\left(S\right)}=\mathit{exp}\left\{-7.94+7.35R{e}^{-0.0748}\right\}$$ (18) $$f_{w\left(R\right)}=\mathit{exp}\left\{-7.53+8.07{\left(\frac{30a_m}{k_s}\right)}^{-0.100}\right\}$$ (19) $$f_1=exp\left\{-0.0513{\left(\frac{R_e}{2.5\times {10}^5}\right)}^{4.65}\right\}$$ (20) $$f_2=\mathit{exp}\left\{-0.0101{\left(\frac{Re}{R_1}\right)}^{2.06}\right\}$$ (21) $$R_1=25{\left(\frac{a_m}{k_s}\right)}^{1.15}$$ (22) If δ/h ≥ 1 (depth-limited), Equation (13).

, where κ is the Karman constant (=0.4), z₀ is the roughness length, n is Manning’s roughness coefficient, k_s is the equivalent roughness (=30 z₀), f_w is the wave friction coefficient, a_m is the amplitude of water particle motion outside the boundary layer (=U_m/σ, σ = 2π/T, T: the wave period), and Re is the Reynolds number defined by Re = U_m a_m/ν where ν is the kinematic viscosity of the fluid.

Firstly, the friction term is not taken into consideration in Method 1, which results in the well-known Green’s equation. Meanwhile, Methods 2 and 3 are based on typical friction coefficients with constant values, $f=0.002$ and $f=0.02$, respectively, as used by Bretschneider and Reid [33]. It is noted that the definition of the friction coefficient in the present study is based on Equation (7), whereas Bretschneider and Reid [33] used a friction factor $C_f$ which is defined without the factor of 1/2 on the right hand side in Equation (7). Therefore, $f=0.002$ and $f=0.02$ in Table 1 correspond to $C_f=0.001$ and $C_f=0.01$, respectively, according to [33]’s definition. In Methods 4 and 5, the steady friction coefficients are applied based on the logarithmic law and Manning’s roughness coefficient, respectively.

Methods 6 and 7 apply a wave friction coefficient with consideration of depth-limited condition explained later. In Method 6, a wave friction coefficient for a rough-turbulent regime proposed by Tanaka [25] will be utilized without recognition of flow regime under wave motion. It is interesting to note that Equation (15) proposed by Tanaka [25] is similar to that theoretically obtained by Humbyrd [34], for which detailed verification has been made by comparing with laboratory data from various sources. Meanwhile, Method 7 is based on Tanaka and Thu’s [24] full range equation for a wave friction coefficient, which enables automatic recognition of flow regime by Equation (16), in which subscripts “L”, “S”, and “R” indicate the friction coefficient for laminar flow, smooth-turbulent flow, and rough-turbulent flow, respectively, and f₁ and f₂ are the weight functions for smooth interpolation between different flow regimes. Sumer and Fuhrman [35] proposed a similar equation applicable to smooth-turbulent, transitional, and rough-turbulent regimes by introducing a weight function. However, since their expression is given in an implicit form with respect to friction factor, an iterated procedure is required to obtain the final result. In addition, the laminar flow regime is not covered in their equation.

Equation (9) with different friction factor in Table 1 has been solved numerically by using the Runge–Kutta method.

2.3. Depth Limitation

According to Tinh and Tanaka [18] and Tanaka et al. [20], the transition from WFZ to SFZ is given by the following equation.

$$\frac{\delta }{h}=1$$

(23)

where $\delta$ is the thickness of the wave boundary layer. After reaching the transitional point between WFZ and SFZ, Equation (13) (logarithmic law) will be applied in the shallower area in Methods 6 and 7, as shown in Table 1.

3. Results and Discussion

3.1. Tsunami Shoaling over Uniform Water Depth

Under uniform water depth condition, Equation (9) can be simplified by substituting $dh/dx=0$ into Equation (9).

$$\frac{dH}{dx}=-\frac{f{H}_{}^2}{3\pi h^2}$$

(24)

Assuming a constant friction coefficient $f$ in Equation (24), the following analytical solution can be obtained [30,33].

$$\frac{H}{H_0}=\frac{1}{1+\frac{f{H}_0}{3\pi h^2}x}$$

(25)

Figure 2 shows a comparison between the analytical solution, Equation (25), and the numerical result from Equation (24) using the Runge–Kutta method, in which the dimensionless quantities defined as $H^{*}=H/H_0$ and $x^{*}=f{H}_{0}x/\left(3\pi h^2\right)$ are utilized where $H_0$ is the wave height at $x=0$. It is confirmed that the numerical analysis has sufficient accuracy as compared with the analytical solution.

3.2. Tsunami Shoaling over Uniform Slope Bathymetry

The second analysis has been made for tsunami propagation over a uniform slope bathymetry.

Table 2. Calculation conditions.

Bottom slope	1/100
Tsunami source depth	h0 = 4000 m
Source tsunami height	H0 = 1 m
Wave period	T = 15 min
Diameter of bed material	d = 0.3 mm

indicates the sea bottom slope and the boundary condition at the tsunami source area, in which h₀ and H₀ are the depth and wave height at tsunami source point (x = 0), respectively. The wave period of tsunami is shown by T, and the representative bottom sediment size is d. Here, it is assumed that Nikuradse’s roughness $k_s$ can be calculated by $k_s=2d$.

Figure 3a shows bathymetry from the source to the shallow area, and Figure 3b indicates computation results for the wave height based on seven different friction factors shown in Table 1. According to this diagram, the differences among the seven methods are invisible in Figure 3b. For this reason, more exaggerated diagrams in the extremely shallow area will be shown later.

In Figure 3c, cross-shore variation of friction coefficient is illustrated. Although there is a distinct difference for friction factor among the seven methods, it does not affect the wave height distribution shown in Figure 3b.

In Figure 3d, the wave boundary layer thickness $\delta$ is obtained from a full-range equation proposed by Tanaka et al. [36], which is an improved version as compared with the original equation [37,38]. It is clearly observed that the ratio of $\delta /h$ is much smaller than 1.0 except for the nearshore region, which means that in most of the computational domain, wave friction factor must be utilized for tsunami computation, as already concluded by Tinh and Tanaka [18]. Finally, Figure 3e indicates a transition of flow regime based on Tanaka and Thu’s criteria [24]. It is highly interesting to note that the sea bottom boundary in the deep area is located in laminar regime, as already reported by [15,20]. Subsequently, a gradual transition occurs to smooth-turbulent and rough-turbulent regime in the shallow water area.

To make a closer inspection for the transitional behavior in Method 7, diagrams for flow regime, wave friction factor, and boundary layer thickness will be illustrated. The transition of flow regime is shown in Figure 4. The boundary layer remains in laminar regime from $h=4000 \mathrm{m}$ to $h=2000 \mathrm{m}$, as already reported by [15,20].

In Figure 5, transitional behavior can be recognized in terms of friction factor from laminar to smooth-turbulent and finally rough-turbulent regime.

A similar diagram can be drawn for wave boundary layer thickness in Figure 6, in which transitional behavior similar to Figure 5 is observed. In Figure 5 and Figure 6, the results of shoaling process are plotted up to the water depth of $h=5 \mathrm{m}$, where another transition occurs from WFZ to SFZ (see Figure 1), as will be explained later.

Figure 7 shows an exaggerated diagram in the shallow area less than $h=20 \mathrm{m}$ over the bathymetry shown in Figure 7a. From Figure 7b, the difference of the wave height is clearly observed dependent on the friction coefficient used in Equation (9).

Figure 7c indicates spatial variation of $\delta /h$ during the shoaling process. Equation (23) is satisfied at $h=5.3 \mathrm{m}$, where the transition from WFZ to SFZ occurs. Therefore, slight discontinuity of friction factor can be seen in Methods 6 and 7 in Figure 7d, due to shifting from wave friction factor to steady flow friction factor. It is interesting to note that the transition occurs at such a shallow place. This result shows good agreement with a field observation during the 2010 Chilean Tsunami by Lacy et al. [38].

A common method used in previous studies on tsunami propagation and corresponding sediment transport based on the Manning formula (Method 5) shows overestimation of the friction factor, whereas steady friction factor from the logarithmic law (Method 4) gives underestimation.

The steady flow friction law has commonly been applied in numerical simulations of tsunami (e.g., Imamura [39], Liu et al. [40]), simply assuming that the period of tsunami is long enough as compared with wind-generated waves. According to the present study, however, when we evaluate wave height attenuation and sediment movement during tsunami propagation, precise identification of such transitional processes in the bottom boundary layer is highly required as employed in Method 7.

4. Conclusions

This study investigates the damping of tsunami height due to bottom friction under shoaling process. As compared with previous studies of Williams and Fuhrman [15] and Tanaka et al. [20], in which Green’s law had been employed, the wave shoaling process has been obtained numerically from an energy flux method. As the wave propagates from the tsunami source area in the onshore direction, first transition occurs in the wave boundary layer from laminar to smooth-turbulent, and finally to rough-turbulent regime. Subsequently, when the depth-limited condition is satisfied, the second transition is observed from WFZ to SFZ.