Run-Up of a Vortex Hydrodynamic Bore onto the Shore

Abstract

The run-up of a vortex hydrodynamic bore onto an inclined beach is the subject of this study. To theoretically analyze this problem, we use the Benny equations, which, within the shallow water model, allow us to take into account the distribution of horizontal fluid velocity along the depth of the fluid layer. We show that the presence of a shear flow behind a bore significantly modifies the different regimes of bore motion toward the shoreline depending on its strength. The subsequent collapse of the bore near the shoreline with the release of a high-speed run-up jet onto the dry shore is also significantly modulated by the degree of vorticity of the fluid flow. The maximum flooding length and run-up height increase significantly with increasing vorticity of the fluid flow. We use a theoretical model based on the characteristic Whitham rule for a bore, supplemented by the laws of conservation of mass and the momentum of a liquid crossing a shock wave. It is assumed that the wave run-up that appears after the “collapse” of the bore is determined by gravity. As a result, the maximum value of the wave run-up, its speed, the influence of flow vorticity, and its structure as a whole are estimated. The acceptable agreement of the simulation results with experimental data can serve as a justification for the applicability of our model to the calculation of the bore run-up onto a sloping beach.

1. Introduction

Characterizing waves approaching the shore and estimating their maximum run-up is one of the most significant tasks in coastal engineering because it is closely related to inundation, property damage, and overtopping [1,2,3,4]. Maximum wave run-up is an important design criterion for various types of coastal structures such as embankments, breakwaters, and seawalls. In addition, beach processes such as dune erosion and storm flooding are related to wave run-up. A correct assessment of the maximum wave run-up can lead to a more economical design. The expected wave run-up determines the height of the crest of a coastal structure designed to prevent wave overtopping. Overestimating the maximum wave run-up can result in significant costs from installing a crushed stone embankment breakwater.

Traditionally, studies have been conducted for periodic waves running ashore, solitary waves of the soliton type, and hydrodynamic bores, all of which simulate a tsunami wave. In turn, incoming waves of various amplitudes are considered, including both non-breaking waves of small amplitude and powerful incident waves with a high steepness and amplitude, and it is the breaking of these waves which is of great practical interest. Nevertheless, an adequate theoretical description of breaking waves still remains elusive in wave theory due to the difficulty of quantitatively describing energy dissipation during wave breaking.

Breaking waves rushing ashore can inundate coastal areas and cause severe coastal flooding, beach erosion, major property damage, and loss of life [1,3,5,6,7,8]. The ocean tsunami waves generated by underwater earthquakes usually have long wavelengths and small wave heights. When waves approach the shore, the wave heights can increase significantly due to bathymetric changes. This is why tsunami run-up on horizontal coastlines is similar to the dam-breaking flow that generates a strong shock of the water level, causing a hydrodynamic bore [9].

Bores in general are quite ubiquitous in nature. Tidal bores and bore-like tsunami waves have been often observed in coastal areas, where the bore front is either undulating or breaking. Tidal bores are often found at the mouths of a number of large, deep rivers. The approach and run-up of bores to the shore and the accompanying tidal currents have been the subject of study in recent decades. Whitham [10] and Keller et al. [11], using the classical shallow water model, theoretically described the evolution of a uniform bore on a planar beach by the method of characteristics. Whitham [10] proposed a mathematical method for approximating the run-up of a hydrodynamic bore onto an inclined beach, defined as a characteristic rule. The speed of fluid flow behind the bore and the speed of the propagation of the bore front approach each other near the shoreline. Shen and Meyer [12] confirmed the correctness of Whitham’s characteristic rule [10] and further described wave motions close to the shoreline. Maximum run-up height and horizontal inundation distance were plotted as functions of initial bore velocity and beach slope. They adopted a model where the wave run-up that occurs after the “collapse” of the bore on a dry beach is determined only by the force of gravity. Peregrine and Williams [13] continued the work of Shen and Meyer [12] and described water flow rates near the shoreline.

In experimental studies investigating the propagation of hydrodynamic bores and accompanying wave fluid flows, various methods of their formation were used [14]. Miller [15] used a piston-type wavemaker to generate undulating and breaking bores. He proposed approximate formulas for the maximum run-up height depending on the height of the bore, the slope of the beach, and the roughness of the beach. A study of wave flows caused by solitons and single hydrodynamic bores was carried out by Baldock et al. [16]. Pujara et al. [17] used a piston-type wave maker to generate several types of waves, such as a solitary wave, successive solitary waves, and a short undulating bore. Their experimental results show that the maximum run-up heights among the wave types are almost the same. At the same time, it was found that successive solitary waves create greater flooding depths compared with single waves. Using a long-stroke wave maker, Barranco and Liu [18] studied undulating and breaking bores of varying strength and length generated in a wave flume. Both non-decaying and decaying bores were investigated. Swash flow characteristics, including maximum inundation depth and run-up height, were measured and analyzed.

The most widely used bore formation mechanism in laboratory experiments is the dam-break system. The initial stages of dam-break flows were studied by Stansby et al. [19] who concluded that the shallow water model is adequate for describing the free surface profiles of breaking bores. An experimental study of undulating and breaking bores on a sloping beach was carried out by Yeh et al. [20]. A stronger slowdown in bore run-up was discovered compared with theoretical estimates based on the classical shallow water theory. Janosi et al. [21] presented the results of experiments on the propagation of breaking bores of varying intensities. Undulating bores on a horizontal beach with different reservoir lengths were investigated by Lin et al. [22] and Lin et al. [23]. They also measured the velocity fields behind the bore. Barranco and Liu [18] studied non-decaying bores that were generated in a laboratory using a dam-break system. The results of their experiments showed the dependence of the depth of flooding, the run-up height, and flood duration on the length of the reservoir and the strength of the bore at the beach toe. Using a particle image velocimetry system, Hornung et al. [24] measured flow velocities under a dam-break generated by a breaking bore at a constant depth. They observed large horizontal velocities near the breaking bore front, which decreased to a constant in the water column. The fluctuating velocities were found to be strongest near the breaking front. Fluctuating velocities were also observed in the entire water column during the bore plateau and the bore tail. They also discussed vorticity measurements and their generation due to unsteady and air entrainment.

Breaking waves are capable of creating significant vorticity in the fluid flow. The phenomenon of vorticity can be caused by the viscosity or the turbulence of the flow even before the arrival of the hydrodynamic bore. Shear flow near the surface can also be caused by onshore or offshore winds. A surface shear wave generated in fast-flowing reverse currents can reach significant heights before collapsing in the coastal zone.

Shugan et al. [25] presented an experimental and theoretical study on the dam-break flow for a flat horizontal bottom. Based on Benny’s shallow water equations, a theoretical model of dam-break flow was constructed. Benny’s equations are an extension of the classical shallow water equations as they make it possible to describe vortex movements in a layer of liquid. This seems to be an extremely important property when describing the incursion of a hydrodynamic bore onto the shore, which, according to our experiments, is accompanied by the formation of a chain of vortices.

We modified the conditions of the shock wave in the presence of wave breaking and the generation of a precursor due to the forward-directed jet. Formation criteria were determined, and the propagation regimes of both breaking and non-breaking hydrodynamic bores were analyzed. A breaking bore is characterized by the formation of a mushroom-shaped jet that consists of two oppositely directed vortices and a forward-directed jet. Together, they form a precursor of the bore and cause spreading at a much higher speed. Accordingly, the speed of the precursor of bore propagation in the form of a jet leads to significantly higher values of the run-up speed of the hydrodynamic bore compared with generally accepted values. The experiments we conducted showed their acceptable correspondence to the theoretical model of dam breaking and further propagation of a hydrodynamic bore. The results of our experiments are in good agreement with the theoretical predictions of the proposed model of dam-break flow and hydrodynamic bore propagation in a wide range of initial conditions of the problem.

As a further development of our previous work, the run-up of a vortex hydrodynamic bore on an inclined plane beach is the main goal of the present research. The use of Benny’s equations makes it possible to take into account the vortex structure of the fluid flow. This problem has a rich set of various hydrodynamic effects, such as “bore collapse” and the generation of a bore precursor in the form of a leading jet, etc. As a result, the maximum value of the wave run-up, its speed, the influence of flow vorticity, and its structure as a whole are estimated. The obtained modeling results may significantly change the generally accepted values of the speed at which tsunami-type waves run-up onto the shore.

2. Hydrodynamics Bore Model

We use a model of the surface wave motion of an ideal constant-density fluid with a free surface. The problem is considered in a Cartesian two-dimensional rectangular coordinate system, with the horizontal x-axis at the level of the quiescent water directed toward the beach and the vertical y-axis directed vertically upward. The sloping bottom is defined as $y=-H_b(x)$ so that the function $H_b(x)$ expresses the variable depth of calm water. The elevation of the free surface is given by $y=\mathsf{\eta }(x,t)$, where $t$ is time and $H_0$ is the initial depth of the water in the horizontal-bottom region (Figure 1). The horizontal speed and the total water depth are denoted as $u(x,y,t)$ and $H=\mathsf{\eta }+H_b$, respectively.

The system of shallow water equations in the form of Benny [26] is as follows:

$$\left\{\begin{array}{l}{u}_t+u{u}_x-{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \underset{-H_B(x)}{\overset{y}{\int }}udy}\right)u_y+g{\mathsf{\eta }}_x=0,\\ {\mathsf{\eta }}_t+{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \underset{-H_B(x)}{\overset{\mathsf{\eta }}{\int }}udy}\right)=0,\end{array}\right.$$

(1)

where $g$ is the gravity acceleration.

Dimensionless variables are introduced as

$$\begin{array}{l}{x}^{’}={\displaystyle \frac{x}{H_0}}; {\mathrm{y}}^{’}={\displaystyle \frac{y}{H_0}}; t^{’}=t\sqrt{{\displaystyle \frac{g}{H_0}};}\\ u^{’}={\displaystyle \frac{u}{\sqrt{g{H}_0}}}; {\mathsf{\eta }}^{’}={\displaystyle \frac{\mathsf{\eta }}{H_0}}; {H_b}^{’}={\displaystyle \frac{H_b}{H_0}}.\end{array}$$

Then, the system (1) takes the no dimensional form

$$\left\{\begin{array}{l}{u}_t+u{u}_x-{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \underset{-H_B(x)}{\overset{y}{\int }}udy}\right)u_y+{\mathsf{\eta }}_x=0,\\ {\mathsf{\eta }}_t+{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \underset{-H_B(x)}{\overset{\mathsf{\eta }}{\int }}udy}\right)=0,\end{array}\right.$$

(2)

where the primes are omitted.

With the additional assumption that the velocity $u$ is a function of only one horizontal coordinate $x$, the system of Equation (2) is reduced to the classical shallow water model:

$$\left\{\begin{array}{l}{u}_t+u{u}_x+{\mathsf{\eta }}_x=0,\\ {\mathsf{\eta }}_t+{\displaystyle \frac{\partial }{\partial x}}\left(u(\mathsf{\eta }+H_B)\right)=0.\end{array}\right.$$

(3)

Solutions to the system of Equation (2) are sought in the following form:

$$\begin{array}{l}u=u_0(x,t)+\mathsf{\alpha }y,\\ \mathsf{\eta }=\mathsf{\eta }(x,t)\end{array}$$

(4)

where $u_0$ and $\mathsf{\eta }$ are functions to be determined, and the free parameter $\mathsf{\alpha }$ specifies the value of the vertical velocity shift, that is, the vorticity of the water flow.

Significant vorticity of the fluid flow can be caused by breaking waves [24]. The breaking developed by the hydrodynamic bore gives every reason to consider the flow behind it as a vortex and to include vorticity as one of the fundamental parameters of the bore motion. Shear flow near the surface can also be caused by onshore or offshore winds [27,28]. An onshore surface shear tends to reduce the height of a wave at breaking; an offshore shear increases it. The phenomenon of vorticity can be caused by the viscosity or turbulence of the flow even before the arrival of the hydrodynamic bore. It can be noted that shear flows with near-linear profiles are usually observed in boundary layers at the free surface of the fluid flow and near the bottom. Nevertheless, in the considered model of wave motion for the shallow water approximation, boundary layers can occupy almost the entire flow region, and it seems quite acceptable for us to consider, as a first approximation, a linear vertical profile of horizontal shear flow. Substituting expression (4) into the system of Equation (2) leads to the following form:

$$\left\{\begin{array}{l}{u}_{0t}+u_0{u}_{0x}+{\mathsf{\alpha }}^2{H}_B{H}_{Bx}-\mathsf{\alpha }{(u_0{H}_B)}_x+{\mathsf{\eta }}_x=0,\\ {\mathsf{\eta }}_t+{\left[u_0(\mathsf{\eta }+H_B)+{\displaystyle \frac{\mathsf{\alpha }({\mathsf{\eta }}^2-H_B^2)}{2}}\right]}_x=0.\end{array}\right.$$

(5)

It should be noted that the Benny equation system (2) is spatially two-dimensional and is written in the coordinate system (x, y). The representation of the solutions to these equations in the form (4), with a linear vertical velocity shift, reduces Equation (2) to one-dimension (5) without any additional assumptions or vertical averaging. Moreover, in our opinion, only a linear velocity shift has this property, and any other approximations for the vertical velocity shift leave the equations of motion two-dimensional and requiring further averaging. Thus, the linear model of vertical velocity shift seems to be very convenient and effective for theoretical analysis.

The conservation equations for mass and momentum are written for one spatial variable and contain the vorticity parameter $\mathsf{\alpha }$, which is set based on the boundary conditions of the problem.

With the following substitution of the functions to be determined as $W=u_0+\mathsf{\alpha }[(\mathsf{\eta }-H_B)/2]=u_0+\mathsf{\alpha }[(H-2H_B)/2]$, $H=\mathsf{\eta }+H_b$, the system of Equation (4) will take the form

$$\left\{\begin{array}{l}{H}_t+{(WH)}_x=0,\\ W_t+{\left({\displaystyle \frac{W^2}{2}}+{\displaystyle \frac{{\mathsf{\alpha }}^2{H}^2}{8}}+H-H_B\right)}_x=0.\end{array}\right.$$

(6)

From the first equation of the conservation of mass of the system (6), it can be seen that the function $W={\displaystyle \frac{1}{H}}{\displaystyle \underset{-H_b}{\overset{\mathsf{\eta }}{\int }}(u_0+\mathsf{\alpha }y)}dy$ (misprint is corrected) represents the average velocity of horizontal fluid flow with a vertical shear.

The equations of motion for shallow water (6) represent a hyperbolic type system for a pair of unknown functions $H(x,t),W(x,t)$ containing a vorticity parameter $\mathsf{\alpha }$.

To describe the vortex bore propagation on an inclined plane beach $H_b(x)=-\tan \mathsf{\beta } x$ (Figure 1), we will follow the approximation method of Whitham [10] called the characteristic rule. The general scheme of the solution method used is as follows. The relevant equations of motion (5) are first presented in a characteristic form:

$$\left(\sqrt{H}\sqrt{1+{\mathsf{\alpha }}^{2}H/4}+W\right)\left(\sqrt{{\displaystyle \frac{1+{\mathsf{\alpha }}^{2}H/4}{H}}}dH+dW\right)-d{H}_b=0$$

(7)

on a forward propagating characteristic, and $dx/dt=W+c\sqrt{1+{\mathsf{\alpha }}^{2}H/4},$ where $c=\sqrt{H}$. Then, the rule is to apply the differential relation, which must be satisfied by the characteristics of the flow quantities just behind the shock wave.

The conservation equations for mass and momentum follow from the vorticity shallow water Equation (6), where the equation for mass is

$$H_t+{(WH)}_x=0$$

(8)

and the equation for momentum is

$${(WH)}_t+{\left(W^{2}H+{\displaystyle \frac{H^2}{2}}+{\displaystyle \frac{{\alpha }^2{H}^3}{12}}\right)}_x=0,$$

(9)

In turn, the hydrodynamics of a bore or a discontinuity of the free surface level propagating with a velocity $U$ is determined using the laws of the conservation of mass and momentum (8,9) of the fluid flow on the shock wave as follows:

$$-U{[H]}_0^1+{[WH]}_0^1=0$$

(10)

$$-U{[WH]}_0^1+{\left[W^{2}H+{\displaystyle \frac{H^2}{2}}+{\displaystyle \frac{{\alpha }^2{H}^3}{12}}\right]}_0^1=0,$$

(11)

where square brackets indicate a jump in functions before 0 shocks and after 1 shock.

Using the initial conditions of calm water ahead of the bore, the conditions at the jump (10,11) can be represented in the following form:

$$\left\{\begin{array}{l}U={\displaystyle \frac{HW}{(H-H_b)}}\\ W=\sqrt{{\displaystyle \frac{(H-H_b)}{H{H}_b}}\left({\displaystyle \frac{H^2-H_b^2}{2}}+{\displaystyle \frac{{\alpha }^2{H}^3}{12}}\right)}\end{array}\right..$$

(12)

By substituting expressions (12) into Equation (7), we can obtain a differential equation for determining the dependence $H=H(H_b,\alpha )$:

$$\begin{array}{l}\left(\sqrt{H}\sqrt{1+{\displaystyle \frac{{\mathsf{\alpha }}^{2}H}{4}}}+\sqrt{{\displaystyle \frac{(H-H_b)}{H{H}_b}}\left({\displaystyle \frac{H^2-H_b^2}{2}}+{\displaystyle \frac{{\mathsf{\alpha }}^2{H}^3}{12}}\right)}\right)\times \\ \times \left({\displaystyle \frac{H(12H_b^3-{\mathsf{\alpha }}^2{H}^4-6H_b^{2}H-6H^3)}{4\sqrt{3}\sqrt{H_b^3{H}^3(H_b-H)(6H_b^2-6H^2-{\mathsf{\alpha }}^2{H}^3)}}}+\left(\sqrt{{\displaystyle \frac{1}{H}}+{\displaystyle \frac{{\mathsf{\alpha }}^2}{4}}}+{\displaystyle \frac{(H_b(12H^3-6H_b{H}^2-6H_b^3)+(3H-2H_b)H^3{\mathsf{\alpha }}^2{H}_b)}{4\sqrt{3}\sqrt{H_b^3{H}^3(H_b-H)(6H_b^2-6H^2-{\mathsf{\alpha }}^2{H}^3)}}}\right)dH\right)-d{H}_b=0\end{array}.$$

(13)

Equation (13) was solved numerically with the initial condition for the strength of the vortex hydrodynamic bore $H=\mathsf{\eta }+1$ in the horizontal-bottom region (Figure 1).

Keller et al. [11] and Ho and Meyer [29] showed that Whitham’s characteristic rule gives the correct solution. The bore height eventually disappears as the bore approaches the shore for any of its strengths.

3. Results

3.1. Height of the Bore Approaching the Shoreline

The change in the height of the bore $\mathsf{\eta }=H-H_b$ when approaching the shoreline is determined by its initial height and the degree of vorticity $\mathsf{\alpha }$ of the flow behind the bore. For a bore that is initially “weak” enough or undular $\mathsf{\eta }\sim (0,0.4)$ [20], it will increase in height, but over time, its height will begin to decrease progressively. At the final stage of approaching the shoreline, the height of the bore tends to zero. The presence of shear flow behind the bore emphasizes this propagation property, and the initial growth of the bore is essentially enhanced. (Figure 2a). For “stronger” shocks $\mathsf{\eta }\sim (0.4,0.7)$, only bores with a shear flow are initially enhanced, while shocks with depth-uniform fluid flow behind the bore lose this dependence and the bore height decreases as it propagates (Figure 2b). The heights of “strong” or fully developed bores $\mathsf{\eta }\ge 0.7$ [20] decrease as they approach the shoreline, finally tending to zero, while the presence of a vorticity flow $\mathsf{\alpha }$ behind the bores weakens this trend (see Figure 2c). In general, it can be noted that the presence of a shear flow behind a bore significantly changes its dynamics, sometimes leading to new propagation effects.

The dynamics of changes in the height of the bore when approaching the shoreline corresponds to the experimental results of Miller [15], Yeh and Ghazali [30,31], and Yeh et al. [20].

3.2. The Speed of the Bore When Approaching the Shoreline and the Run-Up of the Jet on a Dry Beach

The height of a bore tends towards zero near the shoreline. The bore front velocity $U$ initially decreases for all its initial values H then suddenly sharply increases in the vicinity of the shoreline $H_b(0)=0$ and reaches the finite value $U^M$, whereas its accelerations become singular at the shoreline. Thus, the most important parameter that determines the strength of a bore near the shore is the value of $U^M$ (i.e., the final speed at the shoreline, which is also a measure of its initial energy). The dependence of the critical speed $U^M$ on the initial height of the bore H is shown in Figure 3. It can be seen that the maximum velocity $U^M$ increases with an increasing degree of vorticity of the flow behind the bore.

This phenomenon of bore behavior at a shoreline represents the rapid conversion of potential energy to kinetic energy and is often called “bore collapse” (Figure 4a–c).

Shen and Meyer [12] showed that after, the collapse of a bore on a shoreline, further propagation of the front along a dry, sloping shore occurs in the form of a thin jet determined by the simple ballistics of fluid particles under the influence of gravity. This is the so-called run-up process. As such, the run-up front speed $U^R$ can be expressed as

$$U^R={\displaystyle \frac{d{x}_s}{dt}}={\left({(U^M)}^2-2\tan (\mathsf{\beta })x_s\right)}^{1/2},$$

where $x_s$ is the position of the run-up jet front.

The maximum vertical run-up value, accordingly, takes the form

$$R={\displaystyle \frac{{(U^M)}^2}{2}}.$$

From Figure 4, it can be seen that the maximum flooding length and the run-up height increase significantly with increasing vorticity of the fluid flow. It should be noted that experimental studies [20] show significantly lower values of maximum flooding length and run-up height due to the effect of bottom friction.

4. Conclusions

The run-up of a vortex hydrodynamic bore on an inclined plane beach is the main goal of the present research. This problem has a rich set of various hydrodynamic effects, such as a “bore collapse”, the formation of vortices, the generation of a bore precursor in the form of a leading jet, etc. A model based on the use of Benny’s equations made it possible to analyze the influence of fluid flow vorticity on the process of hydrodynamic bore run-up onto an inclined beach. The presence of a shear flow behind a bore significantly modifies the different regimes of bore motion toward the shoreline depending on its strength. The subsequent collapse of the bore near the shoreline with the release of a high-speed run-up jet onto the dry shore is also significantly modulated by the degree of vorticity of the fluid flow. The maximum flooding length and the run-up height increase significantly with increasing vorticity of the fluid flow. We applied Whitham’s characteristic bore rule, together with the laws of the conservation of mass and the momentum of a fluid crossing a shock wave. It was also assumed that the wave run-up that occurs after the “collapse” of the bore is determined by gravity. As a result, the maximum value of the wave run-up, its speed, the influence of flow vorticity, and its structure as a whole are estimated.

An initially weak bore $\mathsf{\eta }\sim (0,0.4)$ increases in height, but over time, its height will begin to decrease progressively. At the final stage of approaching the shoreline, the height of the bore tends to zero. The presence of the shear flow behind the bore emphasizes this propagation property and the initial growth of the bore is enhanced. For a stronger shock $\mathsf{\eta }\sim (0.4,0.7)$, only bores with a shear flow are initially enhanced, while a shock with a depth-uniform fluid flow behind the bore loses this dependence, and the bore height decreases as it propagates. The heights of strong bores $\mathsf{\eta }\ge 0.7$ decrease as they approach the shoreline, finally tending to zero, while the presence of a vorticity flow $\mathsf{\alpha }$ behind the bores weakens this trend.

The most important parameter that determines the strength of the bore near the shore is the value of $U^M$, i.e., the final speed at the shoreline. The bore front velocity $U$ initially decreases for all its initial values H, then suddenly sharply increases in the vicinity of the shoreline $H_b(0)=0$ and reaches the finite value $U^M$. The maximum flow speed $U^M$ increases with increasing initial height of the bore and the degree of vorticity of the flow behind the bore.

After the collapse of the bore, further run-up onto the dry, sloping beach occurs in the form of a thin jet with a much higher speed, determined by the simple ballistics of liquid particles under the influence of gravity. The maximum flooding length and run-up height increase significantly with increasing vorticity of the fluid flow.

Our estimates may significantly change the generally accepted values for the height and speed of a tsunami-like wave run-up onto a beach.