Hydrodynamic Features of an Undular Bore Traveling on a 1:20 Sloping Beach

The hydrodynamic characteristics, including local and convective accelerations as well as pressure gradient in the horizontal direction, of the external stream of an undular bore propagating on a 1:20 sloping beach are experimentally studied. A bore with the water depth ratio of 1.70 was generated downstream of a suddenly lifted gate. A high-speed particle image velocimetry was employed to measure the velocity fields during the run-up and run-down motions. The time series of free surface elevation and velocity field/profile of the generated bore, comprising a pure bore accompanied by a series of dispersive leading waves, are first demonstrated. Based on the fast Fourier transform (FFT) and inverse FFT (IFFT) techniques, the free surface elevation of leading waves and the counterpart of pure bore are acquired separately at a specified measuring section (SMS), together with the uniform horizontal velocity of the pure bore. The effect of leading-wave-induced velocity shift on the velocity profiles of the generated bore are then evaluated at the SMS. To understand the calculation procedure of accelerations and pressure gradient, three tabulated forms are provided as illustrative examples. Accordingly, the relationships among the partially depth-averaged values of the non-dimensional local acceleration, convective acceleration, total acceleration and pressure gradient of the generated/pure bore acquired at the SMS versus the non-dimensional time are elucidated. The trends in the non-dimensional accelerations and pressure gradient of the external stream of generated bore are compared with those of the pure bore. During the run-up motion from the instant of arrival of the bore front to the moment of the peak level at the SMS, continuous decrease in the onshore uniform horizontal velocity, and successive deceleration of the pure bore in the onshore direction are evidenced, exhibiting the pure bore under the adverse pressure gradient with decreasing magnitude. However, the pure bore once ridden by the leading waves is decelerated/accelerated spatially and accelerated/decelerated temporally in the onshore direction during the rising/descending free surface of each leading wave. This fact highlights the effect of pre-passing/post-passing of the leading wave crest on the velocity distribution of generated bore. It is also found that, although the leading waves have minor contribution on the power spectrum of the free surface elevation as compared with that of the pure bore, the leading waves do play an important role on the magnitudes of both accelerations and pressure gradient. The largest magnitude of the acceleration contributed by the leading waves is around 26 times the counterpart contributed by the pure bore. Further, during the run-down motion right after the moment for the peak level of the bore, a linear increase in the magnitude of the offshore uniform horizontal velocity and a constant local acceleration with increasing time are both identified. The partially depth-averaged value of the non-dimensional pressure gradient is equal to a small negative constant (−0.0115) in the offshore direction, indicating that the bore is subject to a constant favorable pressure gradient.


Introduction
Tsunamis are a series of extremely long waves generated by disturbances associated primarily with earthquakes occurring below or near the ocean floor. They can travel across the ocean and cause serious disasters. As indicated by Grue  Most of the damages are related to the run-up and run-down motions of the long waves, which inundates inland and backwashes seawards in the coastal zones. Thus, understanding the features of the long-wave-induced flow field during the whole process is very important for planning and decision making associated with the safeguarding of coastal areas, designs of warning systems, and effective evacuation procedures.
This study, an extension of Lin et al. [2], aims to elucidate the characteristics of the local and convective accelerations as well as the pressure gradient of the external stream of an undular bore, propagating on a 1:20 sloping beach. Previous literatures are briefly reviewed in the following.
With a series of experiments, the flow pattern classification of the bores was reported by Yeh and Ghazali [3] and Yeh et al. [4], dividing into strong and weak bores. A strong bore (or fully developed bore) refers to the condition h 1 /h 0 > 2.0 (where h 1 and h 0 are the upstream and downstream still water depths), while a weak bore (or undular bore) to h 1 /h 0 < 2.0, along with the leading wave breaking for 1.6 < h 1 /h 0 < 2.0. On the other hand, Shimozono [5] reported a classification slightly different from that of Yeh and Ghazali [3] and Yeh et al. [4] with a strong, transitional and weak bore if h 1 /h 0 > 2.0, 1.75 < h 1 /h 0 < 2.0 and h 1 /h 0 ≤ 1.75, respectively.
Parallelly, theoretical and numerical models were developed to study the free surface variations, propagation speeds of bore front, velocity fields and swash zone dynamics for bores traveling over sloping beaches. The 1st-order non-linear long wave theory (Ho and Meyer [14]; Shen and Meyer [15,16]), finite difference method (Packwood [17]), numerical solutions of shallow-water equation ( Guard and Baldock [18]) were also reported. The 2D Reynolds Averaged Navier-Stokes (RANS) solver coupled with k-ε turbulence closure and a volume of fluid method ( Zhang and Liu [19]) as well as a new model (closed-form non-linear evolution equation) derived from the Euler equation ( Kharif and Abid [9]) were subsequently presented.
Experimental studies also illustrated the maximum run-up height (Miller [20]; Yeh [21]), bed shear stress on swash zone (Barnes et al. [22]), and velocity profiles measured by Particle Image Velocimetry (PIV) (O'Donoghue et al. [23]; Kikkert et al. [24]). In the above studies, the leading waves riding on a pure bore were observed but not elucidated. More recently, the pioneering work on the relationship between the free surface elevation and velocity field of the leading waves was presented in Lin et al. [2] using a high-speed PIV (HSPIV). However, the effect of the leading waves on acceleration and pressure gradient of undular bore was not addressed.
As reported by Jensen et al. [25], accelerations are more difficult to obtain by PIV/HSPIV than velocities due to fast spatio-temporal variation in the flow field of solitary waves traveling on a steep beach. To our best knowledge, only two numerical works (Puleo et al. [26]; Higuera et al. [27]) and several PIV/HSPIV or pressure-transducer measurement results ( Chang and Liu [28]; Jensen et al. [25,29]; Pedrozo-Acuna et al. [30]; Lin et al. [31]) have been reported for the solitary/periodic waves. However, no study was undertaken to investigate the characteristics of acceleration and pressure gradient of the bore propagating over a sloping beach. In the present study, the temporal variations of (local, convective, and total) accelerations as well as pressure gradient of the generated undular bore propagating over a 1:20 sloping beach are elucidated in a specified section.
The experimental set-ups and instrumentations are introduced in the next section. Some results and discussions associated with the free surface elevations and velocity fields/profiles are presented in Section 3. The calculation method of accelerations and pressure gradient are detailed in Section 4 with some tables illustrating the calculation process. The temporal variations of accelerations and pressure gradient are then presented in Section 5. Finally, concluding remarks are given in Section 6.

Experimental Set-Ups and Instrumentations
A flume of length 13.0 m, width 0.25 m, and depth 0.5 m was used to perform the bore experiments. A vertical gate, made of steel plate and placed 375.9 cm from the upstream flume end, was opened instantly with the help of a servo-motor mounted on a steel frame structure (see Figure 1a,b). The toe of the 1:20 sloping beach model (made of acrylic sheet) was positioned 270.5 cm downstream from the gate. Before opening the gate, the upstream and downstream water depths correspond to h 1 and h 0 . The sudden opening of the gate thus forms the so-called dambreak-generated bore with the water depth ratio h 1 /h 0 . A Cartesian coordinate system, (x, y), is used with its origin (0, 0) located at the toe of the sloping beach. Positive x is oriented in the horizontal shoreward direction and the positive y in the vertical upward direction over the beach surface ( Figure 1c). The corresponding velocity components of the generated bore (gb) are represented by ([u] gb , [v] gb ). Further, a translated ordinate, y (=y − x/20), is also used to mark the height above the beach surface. Herein, t is time, with t = 0 s denoting the instant for the gate being open initially at x = −270.5 cm. Further, t is time, with t = 0 s identifying the moment when the crest of the 1st leading wave is right at the section where the toe of the sloping beach is located (i.e., x = 0).
The free surface elevations were measured using four capacitance-type wave gauges. The 1st, 2nd, and 3rd wave gauges were placed at x = −150.0, 0, and 101.7 cm, respectively. Note that the still-water-shoreline (SWS) formed by the intersection of the still-water surface and the sloping beach is located at x = x sws = 180.0 cm. The 4th wave gauge was installed at the specified measuring section (SMS) at x = x SMS = 194.0 cm, where the bed was dry prior to arrival of the bore front.
A HSPIV system, with a framing rate ≥ 1000 Hz, was used to measure the 2D instantaneous velocity fields while the bore traveling on the sloping beach. The system comprised an argon-ion laser, a cylindrical lens, and a high-speed digital camera. The 5 Watt argon-ion laser head (Innova-300, Coherent Inc., Santa Clara, CA, USA) was employed as the light source. Titanium dioxide (TiO2) particles having a refractive index of 2. 6 and a mean diameter of 1.8 µm were used as seeding particles. The cylindrical lens was employed to convert the laser beam into a fan-shaped light sheet of about 1.5 mm in thickness. Then the light sheet was diverted upwards via flume glass bottom to illuminate the seeding particles on a vertical plane (which is 10.0 cm from the flume glass sidewall on the camera side). The camera (Phantom M310, Vision Research Inc., Wayne, NJ, USA), with a maximum framing rate of 3260 Hz under the largest resolution of 1280 × 800 pixels, was used to capture the instantaneous particle-laden images. A field of view (FOV), 14.0 cm wide and 8.75 cm high with a resolution of 0.01094 cm/pixel, was centered on the SMS. For calculation of accelerations and pressure gradient of the bore front and leading waves, the camera framing rate was set at 1000 Hz. The sampling interval between two neighboring images is thus ∆t framing = 1.0 × 10 −3 s. The HSPIV algorithm allows one instantaneous velocity field to be determined from a pair of images, starting at the 64 × 64 pixel interrogation window-size and ending at the 8 × 8 pixel. According to Yeh and Ghazali [3], Yeh et al. [4], and Shimozono [5], an undular (or weak) bore occurs downstream of the suddenly lifted gate for h1/h0 ≤ 1.75. For higher value of h1/h0, the flow field induced by the generated bore is more towards transitional state with a certain degree of irregularity. On the other hand, for lower value of h1/h0, the free surface produced downstream of the gate is apt to be a nearly streamlined profile without occurrence of the leading waves (Soares Frazao and Zech [7], and Kharif and Abid [9]). To highlight the role played by the leading waves on modifying the velocity fields/profiles of the pure (undular) bore, the experimental condition with h1/h0 = 1.70 (h1 = 15.3 cm and h0 = 9.0 cm) was eventually selected as the target experiment. Simultaneous measurements of the velocity field and free surface profile/elevation were repeated for 30 runs in the vicinity of the SMS. An ensemble-averaged method followed by a subsequent moving average technique was used to smoothen the velocity fields/profiles and free surface elevations.

Preliminary Tests
At first, a repeatability test for the free surface profile and internal flow structure of the generated bore right downstream of the gate for −268.5 cm ≤ x ≤ −243.5 cm was performed using the camera at a framing rate of 200 Hz. Figure 2a-d shows four images with the internal flow fields underlying the instantaneous free surface profiles, which were randomly selected from 20 repeated runs at the same instants t′ = 0.21 s (or t = −2.464 s). For easily viewing the flow structure, the range of the images is confined only between x = −268.5 cm and x = −258.5 cm. Due to the influence of the initial According to Yeh and Ghazali [3], Yeh et al. [4], and Shimozono [5], an undular (or weak) bore occurs downstream of the suddenly lifted gate for h 1 /h 0 ≤ 1.75. For higher value of h 1 /h 0 , the flow field induced by the generated bore is more towards transitional state with a certain degree of irregularity. On the other hand, for lower value of h 1 /h 0 , the free surface produced downstream of the gate is apt to be a nearly streamlined profile without occurrence of the leading waves (Soares Frazao and Zech [7], and Kharif and Abid [9]). To highlight the role played by the leading waves on modifying the velocity fields/profiles of the pure (undular) bore, the experimental condition with h 1 /h 0 = 1.70 (h 1 = 15.3 cm and h 0 = 9.0 cm) was eventually selected as the target experiment. Simultaneous measurements of the velocity field and free surface profile/elevation were repeated for 30 runs in the vicinity of the SMS. An ensemble-averaged method followed by a subsequent moving average technique was used to smoothen the velocity fields/profiles and free surface elevations.

Preliminary Tests
At first, a repeatability test for the free surface profile and internal flow structure of the generated bore right downstream of the gate for −268.5 cm ≤ x ≤ −243.5 cm was performed using the camera at a framing rate of 200 Hz. Figure 2a-d shows four images with the internal flow fields underlying the instantaneous free surface profiles, which were randomly selected from 20 repeated runs at the same instants t = 0.21 s (or t = −2.464 s). For easily viewing the flow structure, the range of the images is confined only between x = −268.5 cm and x = −258.5 cm. Due to the influence of the initial horizontal and vertical accelerations of water mass beneath the gate, a jet-like flow issues from the upstream side to the shallower quiescent water mass on the downstream side of the gate. horizontal and vertical accelerations of water mass beneath the gate, a jet-like flow issues from the upstream side to the shallower quiescent water mass on the downstream side of the gate. This results in formation of the shear layer and associated vortex structures right downstream of the gate. It is surprisingly found that the flow patterns and core positions of the vortex structures This results in formation of the shear layer and associated vortex structures right downstream of the gate. It is surprisingly found that the flow patterns and core positions of the vortex structures are very similar. This fact highlights lack of prominently irregular fluctuations and high repeatability of the internal flow structure induced by the generated bores for different runs. In addition, the free surface profiles illustrated in the four images, which are summarized in Figure 2e (with those of two additional runs), are almost identical to the ensemble-averaged results of 20 runs. The (relative) value of the global standard deviation for a specified run, which deviates most from the ensemble-averaged result of 20 runs, is only as small as 0.033 cm (0.37% with respect to the downstream still-water depth h 0 = 9.0 cm). All the free-surface evidences also demonstrate high repeatability of the free surface profile of the generated bore right downstream of the gate.
Secondly, a repeatability test for the free surface elevation of the generated bore [η(t)] gb was conducted for 30 repeated runs at x = 0 using a capacitance-type wave gauge. Note that [η(t)] gb is measured with reference to h 0 . As seen in Figure 3a, for clarity and easy comparison, only the results of six runs, randomly selected from the repeated 30 runs, are plotted along with the ensemble-averaged counterpart of 20 runs (which is, in fact, almost same as that of 30 runs due to convergence of the data trend). The generated bore is comprised of a pure bore accompanied by a series of leading waves. It is surprisingly found that any one of the results for six individual runs is almost identical to those of the other five runs. Namely, good agreement is evidenced not only at the crest and trough levels of the leading waves but also at any level of the pure bore without leading wave. The results of six individual runs are also almost same as that of the ensemble-averaged counterpart, only with small differences. The maximum value of the global standard deviation for the specified run, which at most biases the ensemble-averaged result of 30 repeated runs, is 0.041 cm, equivalent to a relative value of 0.8% with reference to the crest level (5.11 cm) of the 1st leading wave at t = 0 s. It is pertinent to mention that the identical trend for the associated results of the free surface elevation obtained at x = 194.0 cm was reported in Lin et al. [2] (Figure 2d on page 79 of reference [2]). Based on the fact abovementioned, the free surface profile/elevation of the generated bore is identified to be produced repeatedly with high repeatability in the present study. Thirdly, Figure 3b presents the time series of the horizontal velocity of the generated bore [u(t)]gb for six individual runs, together with the counterpart of the ensemble-averaged result for 30 repeated runs, measured at the SMS with (x, y′) = (194.0, 1.50) cm. A very similar trend between any two time series and fairly good accordance with the ensemble-averaged result for each of the tested runs are evidenced. The relative value of the global standard deviation is 1.70% with respect to the maximum horizontal velocity (77.01 cm/s) occurring at t = 2.106 s for the crest phase of the 2nd leading wave   Thirdly, Figure 3b presents the time series of the horizontal velocity of the generated bore [u(t)] gb for six individual runs, together with the counterpart of the ensemble-averaged result for 30 repeated runs, measured at the SMS with (x, y ) = (194.0, 1.50) cm. A very similar trend between any two time series and fairly good accordance with the ensemble-averaged result for each of the tested runs are evidenced. The relative value of the global standard deviation is 1.70% with respect to the maximum horizontal velocity (77.01 cm/s) occurring at t = 2.106 s for the crest phase of the 2nd leading wave (due to breaking of the 1st leading wave at x = 101.7 cm, to be stated later). This fact strongly justifies high repeatability of the horizontal velocity of the generated bore propagating on the sloping beach. Figure 4 shows a series of temporal variation in the (ensemble-averaged) free surface elevation of the generated bore, [η(t)] gb and [η (t)] gb , at x = 0, 101. 7 and 194.0 cm, respectively. Note that the still-water surface intersects the slope at x SWS = 180.0 cm and [η (t)] gb is gauged upwards from the beach surface.  For x = 0 and 101.7 cm in Figure 4, at least four prominent leading waves propagate shoreward by the bore with its front reaching x = 0 and 101.7 cm at t = 0 and 0.836 s, respectively. These waves experience the shoaling process during the early run-up stage of generated bore, resulting in an increase in wave height and change in wave period (or frequency), thus confirming the dispersive feature of the leading waves (Lin et al. [2]; Soares Frazao and Zech [7]; Kharif and Abid [9]; Tissier et al. [32]; Bjorkavag and Kalisch [33]). The leading waves eventually break at different positions for distinct times, along with the generated bore reaching the maximum run-up height of 8.86 cm at t = 5.195 s (see Figure 5). The 1st leading wave breaks at the breaking point (bp1) x = xbp1 = 101.7 cm for t = 0.836 s with an air-entrained plunging breaker. For t > 0.836 s and x > 101.7 cm, the bore front is followed by a bubbly turbulent flow, thus leading to energy dissipation of the 1st leading wave. Under such a situation, rise in the free surface elevation of the 1st leading wave no longer exists. Instead, a plateau-shaped free surface with some small-scale oscillations (SSOs) in the time series of [η′(t)]gb is observable after passing of the bore front and prior to arrival of the 2nd leading wave for 1.550 s < t < 2.055 s (see Figure 4 for the SMS at xSMS = 194.0 cm).  For x = 0 and 101.7 cm in Figure 4, at least four prominent leading waves propagate shoreward by the bore with its front reaching x = 0 and 101.7 cm at t = 0 and 0.836 s, respectively. These waves experience the shoaling process during the early run-up stage of generated bore, resulting in an increase in wave height and change in wave period (or frequency), thus confirming the dispersive feature of the leading waves (Lin et al. [2]; Soares Frazao and Zech [7]; Kharif and Abid [9]; Tissier et al. [32]; Bjorkavag and Kalisch [33]). The leading waves eventually break at different positions for distinct times, along with the generated bore reaching the maximum run-up height of 8.86 cm at t = 5.195 s (see Figure 5). The 1st leading wave breaks at the breaking point (bp1) x = x bp1 = 101.7 cm for t = 0.836 s with an air-entrained plunging breaker. For t > 0.836 s and x > 101.7 cm, the bore front is followed by a bubbly turbulent flow, thus leading to energy dissipation of the 1st leading wave. Under such a situation, rise in the free surface elevation of the 1st leading wave no longer exists. Instead, a plateau-shaped free surface with some small-scale oscillations (SSOs) in the time series of [η (t)] gb is observable after passing of the bore front and prior to arrival of the 2nd leading wave for 1.550 s < t < 2.055 s (see Figure 4 for the SMS at x SMS = 194.0 cm).

Free Surface Elevations over Horizontal Bottom and Sloping Beach
al. [32]; Bjorkavag and Kalisch [33]). The leading waves eventually break at different positions for distinct times, along with the generated bore reaching the maximum run-up height of 8.86 cm at t = 5.195 s (see Figure 5). The 1st leading wave breaks at the breaking point (bp1) x = xbp1 = 101.7 cm for t = 0.836 s with an air-entrained plunging breaker. For t > 0.836 s and x > 101.7 cm, the bore front is followed by a bubbly turbulent flow, thus leading to energy dissipation of the 1st leading wave. Under such a situation, rise in the free surface elevation of the 1st leading wave no longer exists. Instead, a plateau-shaped free surface with some small-scale oscillations (SSOs) in the time series of [η′(t)]gb is observable after passing of the bore front and prior to arrival of the 2nd leading wave for 1.550 s < t < 2.055 s (see Figure 4 for the SMS at xSMS = 194.0 cm).     Figure 6a-c, respectively. The data trend demonstrates that the generated bore at the SMS is accelerated and decelerated temporally as well as spatially decelerated and accelerated shoreward for t = 2.093 and 2.119 s. Such a feature is attributable to the effect of pre-/post-passing of the 2nd leading wave crest through the SMS.  Figure 6ac, respectively. The data trend demonstrates that the generated bore at the SMS is accelerated and decelerated temporally as well as spatially decelerated and accelerated shoreward for t = 2.093 and 2.119 s. Such a feature is attributable to the effect of pre-/post-passing of the 2nd leading wave crest through the SMS.  Figure 7a shows the time series of ensemble-averaged data for the combined free surface elevation of the generated bore, η′(t), measured at the SMS for 30 repeated runs. As addressed in Appendix A, the frequency contents in free surface elevation of the leading waves, ranging between 2.00 and 4.20 Hz, are higher than those of the pure bore (from 0.067 to 0.10 Hz) around the SMS. Therefore, using fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) with a frequency demarcation ffd = 0.707 Hz (see Figure A1a,b in Appendix A for the details), the free surface  Figure 7a shows the time series of ensemble-averaged data for the combined free surface elevation of the generated bore, η (t), measured at the SMS for 30 repeated runs. As addressed in Appendix A, the frequency contents in free surface elevation of the leading waves, ranging between 2.00 and 4.20 Hz, are higher than those of the pure bore (from 0.067 to 0.10 Hz) around the SMS. Therefore, using fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) with a frequency demarcation f fd = 0.707 Hz (see Figure A1a,b in Appendix A for the details), the free surface elevation mainly contributed from the 2nd to 5th leading waves, [η (t)] lw , is shown in Figure 7b for the frequency contents greater than f fd . Meanwhile, the free surface elevation provided by the pure bore, [η (t)] pb (=[η (t)] gb − [η (t)] lw ), is obtained for the frequency contents less than f fd = 0.707 Hz, as illustrated in Figure 7a (as marked by dashed line). Given that the 1st leading wave has broken at x = 101.7 cm for t = 0.836 s before reaching the still-water shoreline at x SWS = 180.0 cm (see Figure 5). The remaining four correspond to the 2nd, 3rd, 4th, and 5th leading waves. The instants identified in the time histories of the bore and leading waves by employing distinct vertical lines (each has a circled number) are all tabulated in the sub-table of Figure 7a surface elevation from the trough phase. In addition, a very prominent increase of [uu(t)]gb appears for 2.055 s ≤ t ≤ 2.106 s with [uu]gb = 59.5 cm/s at t = 2.055 s, [uu]gb = 66.7 cm/s for the ascending level at t = 2.069 s, and [uu]gb = 81.7 cm/s for the crest level of the 2nd leading wave at t = 2.106 s (see the circled 3-5 in Figure 7a,b and Figure 8). This indicates that the generated bore is accelerated temporally in the onshore direction during the first rising stage of the 2nd leading wave for 2.055 s ≤ t ≤ 2.106 s. For 2.106 s ≤ t ≤ 2.462 s in Figure 7a,b, the free surface elevation changes from the crest level to the two descending levels at t = 2.150 and 2.211 s with [uu]gb = 78.1 and 61.1 cm/s (see the circled 5-7 in Figure 7a,b and Figure 8). Next, it steps over the zero-down-crossing level at t = 2.248 s with [uu]gb = 56.5 cm/s and then goes further down to the trough level of the 2nd leading wave with [uu]gb = 43.9 cm/s at t = 2.462 s (the circled 7-9). This trend shows that [uu(t)]gb keeps decreasing with time for 2.106 s ≤ t ≤ 2.462 s, implying that the generated bore is temporally decelerated shoreward within this time interval.
Furthermore, for 2.462 s ≤ t ≤ 2.566 s in Figure 7a     In Figure 7a,b for 2.000 s ≤ t ≤ 2.055 s, the temporal variation of [η (t)] lw just reflects the free surface rise from the local trough level of the final SSOs in the broken 1st leading wave at t = 2.008 s (the circled 1), via the ascending stage at t = 2.024 s (the circled 2), then to the ending/zero-up-crossing phase of the 1st/2nd leading wave at t = 2.055 s (the circled 3). The uniform horizontal velocities of the generated bore at these three phases are equal to [u u ] gb = 50.4, 53.4, and 59.5 cm/s (see the circled 1-3 in Figure 7a,b and Figure 8). The trend points out the increase in [u u (t)] gb with increasing free surface elevation from the trough phase. In addition, a very prominent increase of [u u (t)] gb appears for 2.055 s ≤ t ≤ 2.106 s with [u u ] gb = 59.5 cm/s at t = 2.055 s, [u u ] gb = 66.7 cm/s for the ascending level at t = 2.069 s, and [u u ] gb = 81.7 cm/s for the crest level of the 2nd leading wave at t = 2.106 s (see the circled 3-5 in Figure 7a,b and Figure 8). This indicates that the generated bore is accelerated temporally in the onshore direction during the first rising stage of the 2nd leading wave for 2.055 s ≤ t ≤ 2.106 s. The features of temporal variation in the uniform horizontal velocity of generated bore, [uu(t)]gb, during passing of the nth leading wave (with n = 2-4) through the SMS are summarized as (1) successive increase from a certain positive value up to a maximum positive value as the free surface elevation changes from the zero-up-crossing to crest level (see the circled 3-5, 11-12, and 16-17 on the top of Figure 8). (2) Decreasing continuously from a maximum positive value at the crest level, via a certain positive value at the zero-down-crossing level, then down to a positive minimum one at the trough level (see the circled 5-9, 12-15 and 17-19). (3) Consecutive increase from a positive minimum at the trough level to a relatively larger value at the ending level (see the circled 9-11, 15-16, and 19-20). This fact indicates the effect of velocity shift on increasing/decreasing [uu(t)]gb due to continuous ascending/descending of the free surface of the nth leading wave. The first rising/descending free surface of the nth leading wave (with n = 2-4) just corresponds to pre-/postpassing of its wave crest. For easy comprehension, variation trend of the acceleration or deceleration in the uniform horizontal velocity of the generated bore is summarized schematically in Figure 9 with respect to the phases of the leading waves (with n = 2-4).   Figure 8). Next, it steps over the zero-down-crossing level at t = 2.248 s with [u u ] gb = 56.5 cm/s and then goes further down to the trough level of the 2nd leading wave with [u u ] gb = 43.9 cm/s at t = 2.462 s (the circled 7-9). This trend shows that [u u (t)] gb keeps decreasing with time for 2.106 s ≤ t ≤ 2.462 s, implying that the generated bore is temporally decelerated shoreward within this time interval.
Furthermore, for 2.462 s ≤ t ≤ 2.566 s in Figure 7a,b, the free surface elevation varies from the trough level, via the ascending level at t = 2.538 s with [u u ] gb = 47.3 cm/s (see the circled 10 in Figure 7a,b and Figure 8), to the ending/zero-up-crossing level of the 2nd/3rd leading wave at t = 2.566 s with [u u ] gb = 51.5 cm/s (the circled 11). Note that the temporal variations in [u u (t)] gb of the 3rd and 4th leading waves are similar to that of the 2nd leading wave, as evidenced in the serial changes for 2.566 s ≤ t ≤ 3.007 s (the circled 11-16) and 3.007 s ≤ t ≤ 3.462 s (the circled [16][17][18][19][20]. The features of temporal variation in the uniform horizontal velocity of generated bore, [u u (t)] gb , during passing of the nth leading wave (with n = 2-4) through the SMS are summarized as (1) successive increase from a certain positive value up to a maximum positive value as the free surface elevation changes from the zero-up-crossing to crest level (see the circled 3-5, 11-12, and 16-17 on the top of Figure 8). (2) Decreasing continuously from a maximum positive value at the crest level, via a certain positive value at the zero-down-crossing level, then down to a positive minimum one at the trough level (see the circled 5-9, 12-15 and 17-19). (3) Consecutive increase from a positive minimum at the trough level to a relatively larger value at the ending level (see the circled 9-11, 15-16, and 19-20). This fact indicates the effect of velocity shift on increasing/decreasing [u u (t)] gb due to continuous ascending/descending of the free surface of the nth leading wave. The first rising/descending free surface of the nth leading wave (with n = 2-4) just corresponds to pre-/post-passing of its wave crest. For easy comprehension, variation trend of the acceleration or deceleration in the uniform horizontal velocity of the generated bore is summarized schematically in Figure 9 with respect to the phases of the leading waves (with n = 2-4). For t ≥ 3.462 s, the influence of the kth leading waves (with k ≥ 5) on the velocity distribution becomes less significant as compared with those from the 2nd to 4th. As supporting evidence, Figure  10 (1)). This implies that the horizontal velocities contributed by the 5th leading wave at t = 3.462 and 3.526 s are equal to [uu]lw = 0.4 and 1.6 cm/s. Therefore, the relative deviation of the maximum shift of the horizontal velocity influenced by the 5th leading wave is equal to 4.4% with respect to [uu]gb = 36.5 cm/s. Similarly, the relative deviations for the 6th and 7th leading waves are 2.7% and 1.4% (not shown), respectively. The data trend confirms that the free surface variation of the kth leading wave (with k ≥ 5) has a decreasing effect on the velocity shift of generated bore. Consequently, their effect on the velocity distribution during the run-up is thus ignored. In the following, the magnitudes of the horizontal velocity in the profiles at SMS for 3.462 s ≤ t ≤ 6.094 s are considered as being contributed mostly by the pure bore. For t ≥ 3.462 s, the influence of the kth leading waves (with k ≥ 5) on the velocity distribution becomes less significant as compared with those from the 2nd to 4th. As supporting evidence, Figure 10 (1)). This implies that the horizontal velocities contributed by the 5th leading wave at t = 3.462 and 3.526 s are equal to [u u ] lw = 0.4 and 1.6 cm/s. Therefore, the relative deviation of the maximum shift of the horizontal velocity influenced by the 5th leading wave is equal to 4.4% with respect to [u u ] gb = 36.5 cm/s. Similarly, the relative deviations for the 6th and 7th leading waves are 2.7% and 1.4% (not shown), respectively. The data trend confirms that the free surface variation of the kth leading wave (with k ≥ 5) has a decreasing effect on the velocity shift of generated bore. Consequently, their effect on the velocity distribution during the run-up is thus ignored. In the following, the magnitudes of the horizontal velocity in the profiles at SMS for 3.462 s ≤ t ≤ 6.094 s are considered as being contributed mostly by the pure bore. Figure 11 illustrates the temporal variation of the horizontal velocity profile of the generated bore for 3.462 s ≤ t ≤ 6.094 s (also see the circled 20-27 in Figure 7a). As mentioned above, the magnitudes of the velocities in the two profiles at t = 3.462 and 3.526 s (the circled 20 and 21) are nearly the same due to the combined effect of the increasing velocities by passing of the 5th leading-wave crest and the temporally decreasing velocities of the pure bore. An analogous situation is also observed for t = 3.686 and 3.711 s (the circled 22 and 23). However, larger onshore velocities are observed at t = 3.686 s (i.e., corresponding to the trough level of the 5th leading wave) than those at t = 3.711 s (i.e., the ending/zero-up-crossing phase of the 5th/6th leading wave). This is because the major decrease in the horizontal velocity of the pure bore accompanied by the minor increase in the horizontal velocity of the 5th leading wave during the second rising stage for 3.686 s ≤ t ≤ 3.711 s. Afterwards, the uniform horizontal velocity of the generated (or pure) bore decreases continuously from [u u ] gb ≈ [u u ] pb = 23.0 cm/s at t = 4.30 s (the circled 24) to [u u ] gb = [u u ] pb = 0 at t = 6.094 s (the circled 27). Note that flow reversal takes place close to the beach surface for t > 4.822 s and the velocity profile at t = 6.094 s is characterized by the uniform horizontal velocity exactly equal to zero in the middle portion. Moreover, occurrence of the shoreward stream near the free surface and the offshore stream of flow reversal close to the beach surface is clearly evidenced.  Figure 11 illustrates the temporal variation of the horizontal velocity profile of the generated bore for 3.462 s ≤ t ≤ 6.094 s (also see the circled 20-27 in Figure 7a). As mentioned above, the magnitudes of the velocities in the two profiles at t = 3.462 and 3.526 s (the circled 20 and 21) are nearly the same due to the combined effect of the increasing velocities by passing of the 5th leadingwave crest and the temporally decreasing velocities of the pure bore. An analogous situation is also observed for t = 3.686 and 3.711 s (the circled 22 and 23). However, larger onshore velocities are observed at t = 3.686 s (i.e., corresponding to the trough level of the 5th leading wave) than those at t = 3.711 s (i.e., the ending/zero-up-crossing phase of the 5th/6th leading wave). This is because the major decrease in the horizontal velocity of the pure bore accompanied by the minor increase in the horizontal velocity of the 5th leading wave during the second rising stage for 3.686 s ≤ t ≤ 3.711 s. Afterwards, the uniform horizontal velocity of the generated (or pure) bore decreases continuously from [uu]gb ≈ [uu]pb = 23.0 cm/s at t = 4.30 s (the circled 24) to [uu]gb = [uu]pb = 0 at t = 6.094 s (the circled 27). Note that flow reversal takes place close to the beach surface for t > 4.822 s and the velocity profile at t = 6.094 s is characterized by the uniform horizontal velocity exactly equal to zero in the middle portion. Moreover, occurrence of the shoreward stream near the free surface and the offshore stream of flow reversal close to the beach surface is clearly evidenced.

Velocity Fields/Profiles during Run-Down
Motion for 6.094 s ≤ t < 12.00 at SMS Figure 12 presents temporal variation of the horizontal velocity profile of pure bore at the SMS for 6.094 s ≤ t < 12.00 s. The phases of the velocity profiles correspond to those of the free surface elevation shown in Figure 7a,b (see the circled [27][28][29][30][31][32][33][34]. The uniform horizontal velocities [u u (t)] pb are all found to take negative values and increase successively in their magnitudes with increasing time t. For example, [u u (t)] pb = 0, −21.6, −44.4, −62.5, and −63.6 cm/s at t = 6.094, 8.00, 10.00, 11.35, and 11.75 s, respectively. The data trend highlights the positive/negative "local acceleration" of the external stream of the pure bore in the offshore/onshore direction during the run-down at the SMS. In summary, the temporal variations in the free surface elevation and horizontal velocity profile of pure bore with acceleration or deceleration feature throughout the full water depth are shown in Figure 13. The effect of velocity shift is neglected at all of the phases marked by the red solid points (which are located at or very close to the zero-up-crossings of the leading waves). Therefore, the magnitudes of the onshore/offshore uniform horizontal velocities in the profiles of pure bore are justified to decrease/increase continuously for 2.000 s ≤ t ≤ 6.094 s/6.094 s ≤ t < 12.00 s during run-up/run-down motion. The result also exhibits that the pure bore is decelerated/accelerated temporally in the onshore/offshore direction for both intervals. It is pertinent to indicate the difference in the acceleration or deceleration feature between the pure bore and the leading waves, as shown clearly in Figures 9 and 13, respectively.   Figure 14b. The time series of horizontal velocity of the "pure bore" at y = 1.50 cm are almost identical to those obtained at y = 2.00 cm. This fact still holds true for the remaining heights away from the boundary layer, clearly indicating the horizontal velocity profile of pure bore is fairly uniform (except that within the boundary layer). At around the zero-up-crossing and zero-down-crossing levels of the free surface elevation of the nth leading wave (with n = 2-4), the horizontal velocity contributed only by the leading waves is equal or close to 0. This fact allows the horizontal velocity, obtained at a specified height and contributed only by the pure bore, to be estimated at the SMS.
Based on the data points located on or close to the dashed line shown in Figure 14b, the onshore uniform horizontal velocity of the pure bore, [u u (t)] pb , during the run-up is estimated by the exponential decay function (Lin et al. namely, for the run-up motion of pure bore, Equation (1) confirms the temporally successive decrease in the onshore uniform horizontal velocity and continuous shoreward deceleration (i.e., with the negative "local acceleration"). Nevertheless, the pure bore while ridden by the leading waves is decelerated/accelerated spatially and accelerated/decelerated temporally in the onshore direction during the ascending/descending free surface of the nth leading wave (with n = 2-4). The result demonstrates again the effect of pre-/post-passing of the leading wave crest on the velocity distribution of the generated bore. During run-down for 6.094 s ≤ t < 12.00 s, the generated bore, behaving as a pure bore without the leading waves, moves offshore at the SMS. The time histories of free surface elevation and horizontal velocity profile of the pure bore, [η (t)] pb (=[η (t)] gb ) and [u u (t)] pb (=[u u (t)] gb ), are illustrated in Figure 14a,b. Further, the temporal variation of the offshore uniform horizontal velocity during run-down is illustrated in Figure 15 with its trend expressed by a regressed linear equation: [u u (t)] pb = −11.242t + 68.508 for 6.094 s ≤ t < 12.00 s.
Based on the data points located on or close to the dashed line shown in Figure 14b, the onshore uniform horizontal velocity of the pure bore, [uu(t)]pb, during the run-up is estimated by the exponential decay function (Lin et al. [2]; O'Donoghue et al. [23]; Kikkert et al. [24] namely, for the run-up motion of pure bore, Equation (1) confirms the temporally successive decrease in the onshore uniform horizontal velocity and continuous shoreward deceleration (i.e., with the negative "local acceleration"). Nevertheless, the pure bore while ridden by the leading waves is decelerated/accelerated spatially and accelerated/decelerated temporally in the onshore direction during the ascending/descending free surface of the nth leading wave (with n = 2-4). The result demonstrates again the effect of pre-/post-passing of the leading wave crest on the velocity distribution of the generated bore.  During run-down for 6.094 s ≤ t < 12.00 s, the generated bore, behaving as a pure bore without the leading waves, moves offshore at the SMS. The time histories of free surface elevation and horizontal velocity profile of the pure bore, [η′(t)]pb (=[η′(t)]gb) and [uu(t)]pb (=[uu(t)]gb), are illustrated in Figure 14a,b. Further, the temporal variation of the offshore uniform horizontal velocity during run-down is illustrated in Figure 15 with its trend expressed by a regressed linear equation: [uu(t)]pb = −11.242t + 68.508 for 6.094 s ≤ t < 12.00 s.
The linear increase in the magnitude of the offshore uniform horizontal velocity with increasing t is highlighted in Equation (2). The fact indicates the acceleration/deceleration of the external stream of the pure bore in the offshore/onshore direction during the run-down.

Definitions Based on Navier-Stokes Equation
Computation of accelerations and pressure gradient of the external stream of the generated bore The linear increase in the magnitude of the offshore uniform horizontal velocity with increasing t is highlighted in Equation (2). The fact indicates the acceleration/deceleration of the external stream of the pure bore in the offshore/onshore direction during the run-down.

Definitions Based on Navier-Stokes Equation
Computation of accelerations and pressure gradient of the external stream of the generated bore in the x direction is based on the horizontal component of the Navier-Stokes equation (with the positive value shoreward): with ρ = the density of water, µ = the dynamic viscosity of water, and h p = the gravitational force potential. The gravity term in Equation (3) is equal to 0. With long wave and neglecting the boundary layer effect, [∂(∂u/∂x)/∂x] gb and [∂(∂u/∂y)/∂y] gb should be fairly small (to be further stated later). Therefore, Equation (3) is rewritten as: where x, y, t) × ∂u(x, y, t)/∂y] gb ), and [A t (x, y, t)] gb are the local, convective, and total accelerations in the horizontal direction, respectively. From Equation (4), is obtained accordingly. It is pertinent to mention that the HSPIV sampling frequency is 1000 Hz, resulting in a time lapse of ∆t framing = 1.0 × 10 −3 s between any two consecutive images. The tests are carried out by trial-and-error, employing distinct time intervals ∆t and spatial intervals ∆x to calculate the local and 1st convective accelerations of the external stream of the generated bore, [A l (x, y, t)] gb and [A c1 (x, y, t)] gb . However, for computing the 2nd convective acceleration [A c2 (x, y, t)] gb , only one spatial interval in the y direction is used with ∆y = 0.0873 cm. Obtaining reasonable accelerations is more difficult than velocities because of fast spatio-temporal fluctuations even in the fine PIV/HSPIV measurements (Jensen et al. [25]). To find out representative results, the partially depth-averaged method (with the velocities obtained outside/inside the boundary layer are included/excluded) is used to calculate the local and 1st convective accelerations. The influences of distinct time and spatial intervals on the partially depth-averaged (pda) values of both {[A l (x, y, t)] gb } pda and {[A c1 (x, y, t)] gb } pda are evaluated as below.
For a given measuring section (e.g., at the SMS), the relationship among the partially depth-averaged values of the local acceleration {[A l (t)] gb } pda , the 1st and 2nd convective accelerations {[A c1 (t)] gb } pda and {[A c2 (t)] gb } pda , the total acceleration {[A t (t)] gb } pda as well as the pressure gradient {[P(t)] gb } pda of the external stream of the generated bore in the horizontal direction can be expressed as:

Calculation Example for Local Acceleration
In the following calculations, five ∆t intervals are tested by trial-and-error using the central difference scheme to ensure the reasonable values of the local accelerations. Due to rapid ascending or descending of the free surface, the forward or backward difference scheme is used for calculation near the free surface.
The values of y (=y − x/20) as well as the data sets for three horizontal velocity profiles of the generated bore [u(y )] gb , obtained at the SMS (x = x SMS = 194.0 cm) for t = 2.059, 2.069, and 2.079 s (i.e., with ∆t = 0.010 s), are tabulated in order from the leftmost side of Table A1 Table A1 and Figure A2a) become large. The examination testifies the appropriateness of using adequate time intervals to reasonably obtain {[Al(t)]gb}pda close to 519.6 cm/s 2 at t = 2.069 s if the criterion for the relative deviation being less than 4% is adopted.
It is worth noting that, at the beginning of the calculation for the acceleration, a convergence test for the calculated results of {[Al(t)]gb}pda at different time intervals Δt was performed but with failure    Table A1 and Figure A2a). The results reveal that the values of {[A l ] gb } pda of the middle three do not differ so much, and that the relative deviations, D r , of {[A l ] gb } pda with reference to the conditional average vary from 1.1% to 3.1%. However, for relatively smaller/greater time interval, say ∆t = 0.005 and 0.015 s, the relative deviations D r equal to 13.3% and 8.1% (see the bottom in the 5th-9th columns of Table A1 and Figure A2a) become large.
The examination testifies the appropriateness of using adequate time intervals to reasonably obtain {[A l (t)] gb } pda close to 519.6 cm/s 2 at t = 2.069 s if the criterion for the relative deviation being less than 4% is adopted.
It is worth noting that, at the beginning of the calculation for the acceleration, a convergence test for the calculated results of {[A l (t)] gb } pda at different time intervals ∆t was performed but with failure in obtaining the convergent value of {[A l (t)] gb } pda . Namely, it is nearly impossible to obtain the optimal ∆t that corresponds to a convergent value of {[A l (t)] gb } pda . This is why the influences of different time intervals on accelerations should be evaluated by trial-and-error. The aim is to find a promising range of the time intervals, within which the relative deviation of each calculation result of {[A l (t)] gb } pda would only vary within 4.0% of the conditional average. Similarly, it is also impossible to obtain the optimal ∆x that corresponds to a convergent value of {[A c1 (t)] gb } pda in the next subsection. For easy understanding, a flow chart that illustrates the calculation procedure to determine a promising value of either {[A l (t)] gb } pda or {[A cl (t)] gb } pda is presented in Figure 17.

Calculation Example for 1st Convective Acceleration at SMS
As seen in Table A2 Table A2 from the 5th to 10th columns and Figure A2b). The average computed from the 2nd to 4th values amounts to −374.0 cm/s 2 and their relative deviations are 3.9%, 2.8%, and 1.1%, respectively, all less than 4%. The small relative deviations indicate promising selection of the three spatial intervals (i.e., Δx = 0.52, 0.70, and 0.96 cm) to compute reasonable values of {[Ac1]gb}pda toward the conditional average of −374.0 cm/s 2 for t = 2.069 s. Nevertheless, the relative deviations for the 1st, 5th, and 6th values stated above are equal to 30.9%, 24.0%, and 30.6%, demonstrating that too smaller/larger spacing interval does result in significant bias (also see the bottom of Table A2 from the 5th to 10th columns and Figure A2b).

Calculation Example for 2nd Convective and Total Accelerations and Pressure Gradient at SMS
Similar to the previous calculation example, the values of heights y′ with Δy = 0.0873 cm and Step 1: Calculate [Al(y')]gb (or [Ac1(y')]gb) at specified time t = ts with an appropriate Δt (or Δx) [e.g., 0.01 s (or 1.0 cm)] Step 2:

Calculation Example for 1st Convective Acceleration at SMS
As seen in Table A2 Table A2 from the 5th to 10th columns and Figure A2b). The average computed from the 2nd to 4th values amounts to −374.0 cm/s 2 and their relative deviations are 3.9%, 2.8%, and 1.1%, respectively, all less than 4%. The small relative deviations indicate promising selection of the three spatial intervals (i.e., ∆x = 0.52, 0.70, and 0.96 cm) to compute reasonable values of {[A c1 ] gb } pda toward the conditional average of −374.0 cm/s 2 for t = 2.069 s. Nevertheless, the relative deviations for the 1st, 5th, and 6th values stated above are equal to 30.9%, 24.0%, and 30.6%, demonstrating that too smaller/larger spacing interval does result in significant bias (also see the bottom of Table A2 from the 5th to 10th columns and Figure A2b).

Calculation Example for 2nd Convective and Total Accelerations and Pressure Gradient at SMS
Similar to the previous calculation example, the values of heights y with ∆y = 0.0873 cm and velocity data, [v(y )] gb and [u(y )] gb , at the SMS are listed for t = 2.069 s from the 1st to 4th columns of Table A3 in Appendix B. } pda which would not have prominent changes due to small variation of the time and spatial intervals. It is found that use of ∆t = 0.008, 0.010, and 0.012 s as well as ∆x = 0.96 cm can acquire reasonable values of accelerations with all relative deviations less than 4.0% with respect to the conditional averages. Further, an examination of the term {[(µ/ρ)∂(∂u/∂x)/∂x + ∂(∂u/∂y)/∂y] gb } pda exhibits the magnitudes of only O(10 −1 ) cm/s 2 , demonstrating its contribution is so small such that its effect can be ignored.

Results and Discussions
In this section, the results of accelerations and pressure gradient are generalized in the dimensionless form. Herein, the downstream still-water depth h 0 and linear wave celerity C 0 (=(gh 0 ) 1/2 ) are used as the representative length and velocity scales, respectively, together with (h 0 /g) 1/2 and g being the representative time and acceleration scales. All of the dimensionless variables are shown with an asterisk superscript. Accordingly, (x*, y*) are the non-dimensional horizontal

Local Acceleration of Pure Bore at SMS
As stated in Section 4.4, the onshore/offshore uniform horizontal velocity of the pure bore at the SMS can be predicted by Equation (1)/(2). The uniform horizontal velocity is originally defined as the partially depth-averaged value of the horizontal velocities ranging within the fairly uniform zone (i.e., excluding the velocities inside the boundary layer flow or internal stream). Accordingly, the non-dimensional uniform horizontal velocity of pure bore [u u *(t*)] pb at the SMS can be described by two following equations: [u u *(t*)] pb = −0.0115t* + 0.7317 for 63.623 ≤ t* < 125.284.
The value of [A l *(t*)] pb should be equal to {[A l *(t*)] pb } pda because of the uniform feature in the external stream of the pure bore.
Based on Equations (8) and (9), Figure 18 shows     As observed in Figure 19a-c for 21.987 ≤ t* ≤ 25.704, the non-dimensional free surface elevation of the 2nd leading wave undergoes the descending down to the trough level. It varies from the crest level at t* = 21.987, via a specified level at t* = 22.447 and the zero-down-crossing level at around t* = 23.470 (circled 8), then to the second specified level at t* = 24.838, and finally to the trough level at t* = 25.704 (circled 9). Correspondingly, the value of {[A l *(t*)] gb } pda changes from 0.0016, via −0.2329 and −0.0970, then to −0.0672, and eventually to 0. The data trend exhibits that prominent deceleration (i.e., {[A l *(t*)] gb } pda < 0 in the "onshore" direction) of the external stream occurs while the free surface post-passing the crest level and pre-passing the trough level of the 2nd leading wave. Herewith, it is pertinent to epitomize that the values of {[A l *(t*)] gb } pda are all negative with magnitudes increasing and decreasing for 21.987 ≤ t* ≤ 22.447 and 22.447 ≤ t* ≤ 25.704, respectively. This results in the external stream of the generated bore being continuously decelerated temporally and, consequently, the non-dimensional uniform horizontal velocity [u u *(t*)] gb decreasing with the non-dimensional time t*. The evidences are also observed from the velocity profiles shown in Figure 8 (see the circled 5-9).

Local Acceleration of Generated Bore at SMS
In addition, the free surface that experiences the second rising stage of the 2nd leading wave for 25.704 ≤ t* ≤ 26.790 is between the trough level and the ending phase/zero-up-crossing level of the 2nd/3rd leading wave. During the same non-dimensional time interval, {[A l *(t*)] gb } pda does increase positively from 0 at t* = 25.704, via 0.0247 at t* = 25.955, then to 0.1544 at t* = 26.790. This trend demonstrates the effect of the second rising in the free surface of the 2nd leading wave on increasing the positive value of {[A l *(t*)] gb } pda . Eventually, this leads to an increase in the magnitude of the non-dimensional uniform horizontal velocity [u u *(t*)] gb for the external stream, see the velocity profiles marked with the circled 9-11 in Figure 8.
As estimated by Equations (1) and (8), during the run-up, the non-dimensional onshore uniform horizontal velocity contributed by the "pure bore" is always decelerated (i.e., with negative value of the non-dimensional local acceleration shoreward) before the free-surface peak level is reached at t* = 63.623. However, due to passing of the 2nd leading wave through the SMS for 21.455 ≤ t* ≤ 26.790, the non-dimensional onshore uniform horizontal velocity of the "generated bore" exhibits drastic variation in the acceleration and deceleration. As confirmed in Section 4.2, the free surface variation of the kth leading wave (with k ≥ 5) has little effect (see Figure 10) on the velocity shift. This allows one to neglect its effect on the velocity distribution of the generated bore during the run-up. Whether such a minor velocity shift has any effect on the local acceleration should be examined.
The positive and negative extrema of {[A l *(t*)] gb } pda for the 5th leading wave are equal to 0.0015 for t* = 36.144 (i.e., t = 3.462 s, at the zero-up-crossing phase, circled 20) and −0.0216 at t* = 38.744 (i.e., t = 3.711 s, at the zero-up-crossing phase, circled 23). These two relative magnitudes with respect to the positive maximum of the 2nd leading wave (0.5403) are as small as 0.3% and 4.0%. For both the 6th and 7th leading waves, all the relative magnitudes are much lower than 2.0% (not shown), indicating that their tiny contribution can be neglected. Accordingly, the non-dimensional local acceleration is justified to be nearly contributed by the pure bore, i.e., {[A l *(t*)] gb } pda ≈ {[A l *(t*)] pb } pda for t* > 36.144.
During the run-up for 36.144 ≤ t* ≤ 63.623, {[A l *(t*)] gb } pda is approximately equal to {[A l *(t*)] pb } pda (see Figure 20a, Equation (11) reconfirms the non-dimensional local acceleration of the generated (or pure) bore with a constant value of 0.0115 in the offshore direction during the run-down for 63.623 ≤ t* < 125.284, demonstrating increase in [u u *(t*)] pb with t*.  Based on the definition of the convective acceleration, the term [u(x, y, t) × ∂u(x, y, t)/∂x] gb , is expected to play a more important role than [v(x, y, t) × ∂u(x, y, t)/∂y] gb , as seen in Tables A2 and A3 in Appendix B. At t* = 21.987, the crest of the 2nd leading wave exactly passes through the SMS (Figure 7). At this instant, the free surface profile with the crest at the SMS has a relatively symmetric form covering the region 21.22 ≤ x* ≤ 21.92 (191.0 cm ≤ x ≤ 197.3 cm) (see Figure 6b). The maximum horizontal velocity profile ([u*(y*)] gb ) max underneath the crest exists at the SMS for t* = 21.987. However, for the crest of the 2nd leading wave pre-passing and post-passing the SMS (see Figure 6a,c), the actual position for the crest and the maximum horizontal velocity profile should appear for x* < 21.56 and t* < 21.987 as well as for x* > 21.56 and t* > 21.987, respectively. This leads to the negative and positive values of the convective acceleration for 20.881 ≤ t* < 21.987 (during the first rising stage) and 20.987 ≤ t* < 25.704 (during the entire descending stage), respectively. The horizontal velocity gradients take negative and positive values for the former and latter intervals, respectively. Further, during the second rising stage for 25.704 ≤ t* < 26.790, the crest of the 3rd leading wave approaches the SMS (at which passing of the crest of the 3rd leading wave occurs at t* = 27.458). This is similar to that illustrated for the 2nd leading wave for t* < 21.987 (i.e., pre-passing of the 3rd leading-wave crest results in [u*∂u*/∂x*] gb < 0), thus rendering the convective acceleration to be negative, as evidenced in Figure 19c (Figure 19c). The external stream of the generated bore, which is subject to the favorable pressure gradient, is accelerated spatially and temporally for 25.704 ≤ t* ≤ 26.790. A complete chronology of variations both in the pressure gradient (from favorable, via zero, to adverse pressure gradient) and in the flow situation (from positive, via zero, to negative local acceleration) from the first rising stage up to the crest phase, via the entire descending stage, then to the second rising stage of the 2nd leading wave is thus outlined.

No Influence of Leading Wave
As seen in Figure 20a,b for the ending/zero-up-crossing phase of the 4th/5th leading wave at t* = 36.144 (t = 3.462 s), the values of {[P*] gb } pda and {[A l *] gb } pda are equivalent to 0.0019 and 0.0015, along with {[A c *] gb } pda = −0.0033. The relative deviations of the magnitude is equal to 0.4% and 0.3% (with respect to the positive maximum of {[A l *] gb } pda = 0.5403 for the 2nd leading wave). The result demonstrates that the two associated values of the 5th leading wave, {[P*] gb } pda and {[A l *] gb } pda , for 36.144 ≤ t ≤ 38.744 are much smaller than those of the 2nd leading wave. Up to now, the remaining kth leading waves (with k ≥ 5) has been demonstrated to play a less or minute role on the velocity shift as well as the contribution to accelerations, thus resulting in the generated bore almost being contributed by the pure bore.
As illustrated in Section 6. From Equations (5), (9), and (11) for 63.623 ≤ t* < 125.284, the partially depth-averaged value of the non-dimensional pressure gradient of the generated (or pure) bore in the onshore direction is obtained: namely, −{[P*]gb}pda = −[P*]pb = −0.0115 is negative in the offshore direction. The fact exhibits that, during the run-down for 63.623 ≤ t* < 125.284, the external stream of the generated bore (or pure bore) is accelerated in the offshore direction under a non-dimensional favorable pressure gradient of −0.0115. This fact is also illustrated in Figure 21a,b, together with the non-dimensional free surface elevation displayed for 36.144 ≤ t* < 125.284. From Equations (5), (9), and (11) for 63.623 ≤ t* < 125.284, the partially depth-averaged value of the non-dimensional pressure gradient of the generated (or pure) bore in the onshore direction is obtained: This fact is also illustrated in Figure 21a,b, together with the non-dimensional free surface elevation displayed for 36.144 ≤ t* < 125.284.

Systematic Summary of Procedure for Exploring Features of Accelerations and Pressure Gradient
To achieve a complete understanding for the features of accelerations and pressure gradients of the generated and pure bores, a systematic summary of the frame structure for different steps is illustrated through the flow chart shown in Figure 22. Note that the relationships among the free surface elevations, velocities, accelerations, and pressure gradients of the pure bore and generated bore are highlighted. The sections/subsections/equations corresponding to a specified working item are also included for reference. It is expected that the flow chart can be used as a guideline for further studies on the undular bores with different values of h 1 /h 0 and propagating on sloping beaches with distinct slopes.

Concluding Remarks
The HSPIV experimental results on the characteristics of both velocity fields and accelerations as well as pressure gradient in the horizontal direction for the external stream of a dambreakgenerated undular bore (with h1/h0 = 1.70), propagating over a 1:20 sloping beach during the run-up and run-down, were presented. Major findings can be drawn as follows:  (6) and (8), respectively. The trends exhibit the successive decrease in the non-dimensional (onshore) uniform horizontal velocity and continuous deceleration of the pure bore in the onshore direction. However, the pure bore accompanied by Measure  Pure bore Generated bore Figure 22. Flow chart systematically illustrating relationships among free surface elevations, velocities, accelerations, and pressure gradients of pure and generated bores.

Concluding Remarks
The HSPIV experimental results on the characteristics of both velocity fields and accelerations as well as pressure gradient in the horizontal direction for the external stream of a dambreak-generated undular bore (with h 1 /h 0 = 1.70), propagating over a 1:20 sloping beach during the run-up and run-down, were presented. Major findings can be drawn as follows: 1.
The 1st leading wave breaks at x* = x bp1 /h 0 = 11.3 for t* = 8.728, leading to the dissipation of energy due to lifting motion of air bubbles under a bubbly turbulent flow. Under such a situation, rise in the free surface elevation of the 1st leading wave no longer exists. Instead, a plateau-shaped pure bore) is accelerated in the offshore direction with a non-dimensional acceleration of 0.0115 and under a non-dimensional favorable pressure gradient of −0.0115.
Finally, two integrated flow charts (Figures 17 and 22), which address the frame structure and relationships among distinct data processing steps, can be applied to a systematic exploration of the undular bores having different values of h 1 /h 0 and propagating on sloping beaches with distinct slopes.

Appendix A Decomposition of Free Surface Elevation Using FFT and IFFT
Using the peak-to-peak or trough-to-trough counting method, the periods and frequencies of the leading waves can be determined directly from the time series of ensemble-averaged free surface elevation of the generated bore [η (t)] gb at the SMS (x = x SMS =194.0 cm). As seen in Figure 4, they vary approximately from 0.25 to 0.50 s and from 2.00 to 4.00 Hz. In addition, the characteristic time and frequency scales of the long-wave-featured pure bore are estimated to be around 10.00-15.00 s and 0.067-0.10 Hz approximately, much lower than those of the leading waves. Note that the time series of the free surface elevation of the generated bore is, in fact, a temporally transient, non-stationary signal. If the temporal variation of frequency underlying such a signal is considered to be the target of interest, it would be better to use Hilbert-Huang Transform or Wavelet Transform for further study. However, the topic is outside the scope of the present study. The main concern herein is to separate the time series of the generated bore into the counterparts of the pure bore and leading waves by taking advantage of their very distinct frequency contents. Under such a situation, it would be convenient to utilize the FFT and IFFT techniques for achieving this goal.
Using FFT, the power spectrum S(f ) of the ensemble-averaged free surface elevation of the generated bore at the SMS is obtained and shown in Figure A1a with a frequency resolution ∆f = 0.0337 Hz. Due to each leading wave behaving like a transient wave but not a periodical wave, absence of prominent spectral peak at the corresponding frequency of each leading wave is clearly evidenced in Figure A1a. Most energy is distributed over relatively low frequency range, say f < (0.606-0.908) Hz, revealing main contribution from the pure bore. As generally expected for a nonlinear free surface disturbance, the higher harmonics and lower subharmonics (Lin [34]) of the pure bore exist in the spectrum for 0.0337 Hz < f ≤ 0.908 Hz approximately (at least with five higher harmonics). In addition, the counterparts for the leading waves range about from 0.606 to 9.032 Hz, thus leading to the combined contribution from the pure bore and leading waves for 0.606 Hz ≤ f ≤ 0.908 Hz. Herein, a frequency demarcation f = f fd = 0.707 Hz is first identified to separate the free surface elevation of generated bore into the counterparts of pure bore for f < f fd and leading waves for f > f fd .
Applying IFFT to the amplitude spectrum of [η (t)] gb with f fd = 0.707 Hz, the time series of free surface elevation for the pure bore are presented in Figure A1b. Only the time series of free surface elevation for 0 s ≤ t ≤ 7.0 s is shown for easy comparison (the counterparts for 7.0 s ≤ t ≤ 13.0 s before and after performing IFFT are totally identical due to absence of the leading waves). It is also interestingly found that, if f fd is designated to be 0.606, 0.740, 0.807, or 0.908 Hz, discernable difference between the corresponding free surface elevation of pure bore and that for f fd = 0.707 Hz is hardly observed in Figure A1b. Especially, the free surface elevations of the pure bore for f fd = 0.707 and 0.740 Hz are almost identical. This is because the ratios of the energy underlying the power spectrum between 0.606 Hz ≤ f ≤ 0.707 (or 0.740) Hz and 0.707 (or 0.740) Hz ≤ f ≤ 0.908 Hz to the counterpart between 0.0337 Hz ≤ f ≤ 0.707 Hz is as small as 0.15% (or 0.23%) and 0.18% (or 0.11%), respectively. The results strongly demonstrate almost no influence on the separation of the free surface elevation not only for 0.707 Hz ≤ f ≤ 0.740 Hz but also for 0.606 Hz ≤ f ≤ 0.908 Hz. However, for f fd ≥ 0.9754 Hz or f fd ≤ 0.539 Hz, more prominent deviation of the free surface elevation of the pure bore takes place with respect to that for f fd = 0.707 (or 0.740) Hz, eventually leading to inappropriate separation of the free surface (not shown).