Linear Analysis of Longshore Currents Instability over Mild Slopes

Longshore current instability is important to nearshore hydrodynamic and sediment transport. This paper investigates the longshore current instability growth model based experimental data with different velocity profiles of slopes 1:100 and 1:40 by adopting a linear shear instability model with the bottom friction effects. The results show that: (1) Only backshear mode exists in the instability of longshore current for slope 1:40 and frontshear and backshear modes may exist slope 1:100. (2) The peaks of linear instability growth mode for slope 1:100 correspond to three cases: the dominant peak is formed by the joint action of both frontshear and backshear, or by backshear alone without the existence of the smaller peak or formed by either the frontshear or backshear. (3) Bottom friction can decrease the corresponding unstable growth rate but it cannot change the unstable fluctuation period. The results of fluctuation period, wavelength and spatial variation obtained by the analysis of linear shear instability are in good agreement with experimental results.


Introduction
When waves approach obliquely toward the coast, they will force longshore current which flow along the coast. Many problems involved in the process, such as wave breaking, wave boundary layer and turbulence, and strong nonlinear fluid motion are still not completely resolved today. Longshore current, being complex in its movement theory, is also one of the main dynamic factors to account for coastal pollutant movement, sediment transport and coastal morphology evolution (Burchard et al., 2008;Zou, 2015, 2016;Chen, 2005). Fleming and Swart (1982) indicated that 10% error of the longshore current prediction may lead to 70% error of longshore sediment transport. Longuet-Higgins and Stewart (1964) proposed the concept of radiation stress to account for the wave-driven currents, and stated that longshore current is mainly caused by radiation stress and uneven distribution of wave setup induced by uneven distribution of wave height. Oltman-Shay et al. (1989) found that left and right swings exist in longshore current velocity vectors, which are finally proved to be an unstable movement of the longshore current. Miles et al. (2002) pointed out that 16% of sediment transported in the crossshore direction and 12% in the alongshore direction are caused by longshore current instability. The study of Sánchez et al. (2003) showed that the longshore current instability played a major role in the formation of coastal rhythmic terrain. All of these studies showed that the longshore current instability had an important effect on sediment transport and shoreline evolution. Since the unsteady movement of longshore currents was firstly observed by Oltman-Shay et al. (1989), considerable attention has been paid to its characteristics. Feddersen et al. (2011) simulated the field diffusion process of HB06 under the action of wave and longshore current at Huntington Beach in California. The simulation results were in good agreement with the low-frequency vortex and mean longshore current observed in the surf zone shown by dye diffusion. By using a linear instability theory, Bowen and Holman (1989) interpreted the periodic low-frequency oscillation observed by Oltman-Shay et al. (1989) as a result of shear wave or longshore current instability motion. They illustrated the mechanism of shear instability through a simple velocity profile. There was only one extremum in the field of the background vorticity on the offshore side of the longshore current. The instability motion was found to be closely related to the extrema of the backshear, and the results were in line with those observed by Oltman-Shay et al. (1989). Reniers and Battjes (1997) showed that the frequency and wave number predicted by the backshear instability model were consistent with the physical measurements.
Most of the above studies were based on the linear analysis of longshore currents instability and the longshore current velocity profile in their consideration had only one extremum in the field of the background vorticity, so it was called backshear. Baquerizo et al. (2001) proved the existence of instabilities due to the presence of a second extremum of the background vorticity at the front side of the longshore current by using an analytical model based on Bowen and Holman (1989), which was closely related to the frontshear. These linear instability problems for longshore currents were formulated for frictionless flow. Through solutions to the linear instability problems with friction, Dodd et al. (1992) compared the instability properties of velocity profiles over flat and sand bar topography, and found that bottom friction provided a plausible mechanism for the damping of the unstable modes. The background vorticity for the sand bar topography had two extrema, one was related to the backshear, and the other to the frontshear. Further study of the applicability of linear instability analysis to the interpretation of oceanic observations was presented in Dodd (1994). There were also some theories that considered the mechanism and characteristics of the longshore current instability based on nonlinear analyses (such as Özkan-Haller and Li, 2003), but their analyses were still based on the results of unstable wavelengths calculated by linear analysis. These studies showed that the low frequency swing of longshore current instability may be caused by the mutual interference between the frontshear and the backshear.
In order to analyze the actions of the frontshear and the backshear, this paper obtains different instability growth modes of slopes 1:100 and 1:40 and illustrates the causes of different growth modes by using the linear shear instability model considering the bottom friction effects. The goal of this article is to find out the decisive factors of the longshore current instability in the experimental condition to better guide the engineering practice.

Governing equations
Assuming that the horizontal flow velocity (u, v) in the flow field is composed of the mean longshore current velocity V(x) and the disturbance velocity (u', v'), and the wave surface elevation η is composed of the averaged wave surface elevation and the disturbance wave surface eleva-tion η', written as: By substituting Eqs.
(1)-(3) into the horizontal two-dimensional nearshore circulation equation, under the assumption that the perturbations u', v' and η' are small, neglecting the nonlinear and dissipation terms, the perturbation governing equation can be derived.
where the bottom friction coefficient μ=(2/π)f cw u wa , u wa is the horizontal velocity amplitude of wave orbital velocity at the near bottom. With the "rigid lid" assumption, by introducing the flow function Ψ that satisfies the continuous equation, it can be derived by subtracting the derivative of Eq. (6) with respect to x from the derivative of Eq. (5) with respect to y.
Assume that Eq. (7) has a solution form of φ where is the amplitude of the stream function, k is the wave number, ω=ω r +iω i , ω r is the circular frequency of the longshore current fluctuation and ω i is the growth rate of the longshore current instability.
By substituting Eq. (8) into Eq. (7), the governing equation of longshore current instability that takes bottom friction into consideration can be derived as follows: in which c=ω/k=c r +ic i , c r =ω r /k is the phase velocity of longshore current instability. Offshore and shore line boundary conditions By solving the eigenvalues c=c r +ic i of the equations under these conditions, the growth rate ω i and the propagation velocity c r of the longshore current instability can be calculated, and then ω=ω r +iω i =kc r +ikc i can be obtained.

Linear instability mode of longshore current
The essential condition for shear instability is the existence of an inflection point in the velocity profile. Fig. 1 shows two typical longshore current velocity profiles, where the shoreline is located at x=x 2 . One inflexion point is located at x=x s on the offshore side of the left figure, therefore the longshore current is unstable; such situation is the backshear instability. In the right figure, in addition to an inflection point at x=x s1 on the offshore side, there is another inflection point at x=x s2 on the nearshore side, and the corresponding longshore current may also be unstable. The former corresponds to the backshear instability while the latter corresponds to the frontshear instability. In the present study, only the above two cases are considered.
The experiment in this paper was carried out in the multifunctional wave basin of the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology with the dimensions of 55 m in length, 34 m in width and 0.7 m in depth. To study the distribution characteristics of longshore current velocity over mild slope topography, the slopes 1:100 and 1:40 were chosen, respectively. In the experiment, the cross-shore distributions of longshore current velocity fields as well as the wave run-up were measured based on the straight beach model with the two different slopes mentioned above. Fig. 2 shows the lateral view of the 1:100 slope beach model (similar to 1:40 slope). Theoretically, for the slope 1:40, there would be only one inflection point on the offshore side of the maximum velocity of longshore current, while for the slope 1:100, there would be one inflection point on both the offshore and shoreward side, respectively. Therefore, based on the linear instability analysis of the mean longshore current velocity profile under the condition of slopes of 1: 100 and 1: 40, the two modes, backshear and frontshear, can be obtained, the backshear appears for slope 1:40, and both the backshear and the frontshear appear for slope 1:100. It can be seen from the figure that when the bottom friction coefficient f cw is 0.0001, the calculated results of corresponding linear instability analysis are similar to those obtained without considering the bottom friction effect. The growth rate trend is similar, but the corresponding unstable growth rate decreases slightly. When the bottom friction coefficient f cw increases to 0.0005, the location of the dominant peak with respect to the unstable wave number k remains almost the same with that of the case without considering the bottom friction effect, while the growth rate disappears for the range of higher wave number. The unstable wave number k corresponding to the first peak decreases slowly and the corresponding propagation velocity c r is almost the same. When the bottom friction coefficient f cw increases to 0.0008, the results of linear instability analysis remain only a single peak corresponding to the dominant peak in the unstable growth mode without considering the bottom friction effect, and the corresponding unstable growth rate is small. When the bottom friction coefficient is further increased to the critical value of f cw =0.001, the unstable growth rate is 0, and the instability disappears at this time. The above result shows that the bottom friction has little effect on the dominant unstable growth mode or the unstable wave number k and the propagation speed c r (equivalently the unstable fluctuation period) corresponding to the dominant unstable growth mode, but it will cause a decrease of the unstable growth rate.

Linear instability analysis of experimental longshore current
For the governing equation (9) of longshore current instability, the unstable growth pattern of the longshore current can be obtained as long as the velocity distribution V(x) and water depth h(x) are given. The spline function is used to fit the experimental mean longshore current velocity profile. The corresponding linear instability growth mode can  SHEN Liang-duo et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 675-682 677 be obtained conveniently by the spline curve. Fig. 4 shows the fitting results of longshore current profiles generated from regular wave (T=1.5 s, H=2.53 cm) and the irregular wave (T=1.5 s, H rms =3.57 cm) for a beach slope 1:100 (displayed by solid line). While Fig. 5 shows the fitting results of longshore current profiles generated from regular wave (T=1.5 s, H=6.50 cm) and the irregular wave (T=1.5 s, H rms =4.49 cm) for a beach slope 1:40 (displayed by solid line). From the above fitting results of velocity profiles, it can be seen that a downward concave trend occurs on the shoreward side of the mean longshore current velocity profile for the beach slope 1:100, while an upper convex trend occurs for the beach slope 1:40. The longshore current instability is dependent on the velocity profile, which varies for different beach slopes 1:100 and 1:40. The effect of different velocity distributions on linear instability of longshore current will be discussed by using the linear instability theory with the friction value f cw =0. Fig. 6 and Fig. 7 show the variations of the growth rate ω i and propagation speed c r of longshore current instability versus wave number k in the case of the above-mentioned regular wave and irregular wave respectively. Fig. 7 shows that there is only a major peak in the growth rate curve of longshore current linear instability for slope 1:40. Since the mean longshore current velocity profile for slope 1:40 has only one inflection point on the offshore side of its maximum value, the major peak of the growth rate curve calculated by the linear instability corresponds to the backshear. Fig. 6 shows that the growth rate of the longshore current linear instability for slope 1:100 includes two cases: one contains two larger peaks and the other contains only one larger peak (the sub-peak is relatively small). Because there is only one inflection point on both   sides of the longshore current velocity profile for slope 1:100, the unstable growth rate calculated from its linear instability may be caused by frontshear, backshear or their combination. In order to illustrate the point that the peak of the growth rate curve calculated by linear instability theory for slope 1:100 is caused by frontshear or backshear, this paper compares the unstable growth rates between the new one calculated by the new velocity profile and the original one calculated by the original velocity profile. To study the influence of frontshear, the new velocity profile is defined by keeping the velocity profile from the maximum value of longshore current towards offshore unchanged and using a straight line instead of its nearshore side velocity distribution. Similarly, to study the influence of backshear, the new velocity profile is defined by keeping the velocity profile from the maximum value of longshore current towards nearshore unchanged and using a straight line instead of its offshore side velocity distribution. The advantage of doing this is to remove unnecessary interference and preserve the maximization of the unaltered parts.
The unstable growth rate curves corresponding to experimental conditions for slope 1:100 are obtained by the above methods. The results show that the peaks of linear instability growth mode for slope 1:100 correspond to three cases: (1) The curve of the unstable growth rate has two major peaks, the dominant peak is induced by the joint action of frontshear and backshear, and the other is induced by backshear or frontshear. (2) The curve of the unstable growth rate has only one major peak, and the other peaks are relatively small. The major peak is induced by the joint action of frontshear and backshear. (3) The curve of the unstable growth rate has only one major peak, and the other peaks are relatively small. The larger peak is only induced by backshear.
Taking the three regular wave cases (T=1.0 s, H=2.52 cm), (T=2.0 s, H=4.80 cm) and (T=1.0 s, H=4.90 cm) of slope 1:100 for examples, the mechanism that the abovementioned peak is generated by frontshear or backshear or their combination will be illustrated. The changes of the linear unstable growth rate curve versus the velocity profiles should be kept a close watch on.
Figs. 8a-8c show the mean longshore current velocity profile of the above three wave cases without frontshear or backshear respectively and the corresponding linear instability growth rate curves. Fig. 8a shows that using the fitted velocity profile in the range from the maximum value point of longshore current towards offshore, while replacing the shoreward part of the velocity profile with a straight line, is equivalent to removing the effect of the velocity inflection point on the shoreward side of the mean longshore current profile, i.e., removing the effect of the frontshear. It is found that the unstable growth rate curve is similar to that ob-  SHEN Liang-duo et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 675-682 tained from the experimental fitting, which has one major peak and one smaller peak. Similarly, using the fitted velocity profile in the range from the maximum value point of longshore current towards shoreward end, while replacing the velocity profile in the offshore side with a straight line, is equivalent to removing the effect of the velocity inflection on mean longshore current on the offshore side, i.e., removing the effect of backshear. The unstable growth rate curve has only one major peak. Therefore, the major peak of the unstable growth rate curve corresponding to the velocity profile obtained from the experimental fitting is caused by the joint-action of frontshear and backshear while the smaller peak is caused by backshear. Fig. 8b shows that the unstable growth rate curve corresponding to the velocity profile obtained from the experimental fitting has only one major peak and the other peaks are relatively small. The curves corresponding to frontshear (without the effect of backshear) and backshear (without the effect of frontshear) also have only one peak and both of them have a great contribution. Therefore, the peak of the unstable growth rate curve corresponding to the velocity profile obtained from the experimental fitting is caused by the joint-action of frontshear and backshear. Fig. 8c shows a similar trend with that shown in Fig. 8b. The curves corresponding to backshear (without the effect of frontshear) coincide with the unstable growth rate curve corresponding to the velocity profile obtained based on the experimental fitting. Meanwhile, as the unstable growth rate corresponding to the frontshear (without the effect of backshear) is small, the peak of the unstable growth rate curve corresponding to the velocity profile obtained from the experimental fitting is believed to be caused by backshear in this case. Table 1 gives a comparison in fluctuation period between the spectrum analysis and the linear instability analysis (corresponding to the first peak) for slope 1:100. Among them, the fluctuation period obtained by spectrum analysis is determined by the curve which has the maximum peak value at the location of the maximum value of the longshore current velocity based on the three curves (three groups of experiments corresponding to the same wave condition). The fluctuation period is calculated by three frequencies corresponding to three larger peaks in the curve (Shen, 2015). By comparison, it is found that the wave period calculated from the first peak is close to that of the dom- inant unstable period obtained from the spectrum analysis, and the maximum relative error is smaller than 25%. It is shown that the longshore current instability in these wave cases belongs to linear instability stage or weak nonlinear instability stage, which can be explained by the linear instability analysis.

Spatial variation characteristics
The distribution characteristics of the flow field of longshore current instability are given in this section, which can be obtained by the superposition of the disturbance velocity field (u', v') and the mean longshore current velocity field V. The detailed process is that after the wave number k 0 corresponding to the maximum growth rate is obtained, the eigenvalue of the corresponding unstable propagation velocity c and its corresponding eigenvector can be calculated. The stream function Ψ(x, y, t) can be consequently expressed by (x)= r +i i and ω=ck=ω r +iω i as follows: The disturbance velocity field can be calculated by Eqs. (12) and (13).
The amplitude of the above disturbance velocity increases exponentially with time. Since the theory is based on the linear instability model, nonlinear factors will play a leading role as time goes on, so that the linear instability theory will not be established. Therefore, Eqs. (12) and (13) φ φ can only give the initial growth mode of the corresponding velocity profile, that is to take t=0. The amplitude of the stream function is normalized in the process of calculation (take r =1 when i =0). It is necessary to point out that the maximum value of the disturbance velocity is about 1/6 of the maximum value of the mean longshore current (Ren et al., 2011). Therefore in the calculation process of the total velocity field (u', V+v'), the value of the disturbance velocity field calculated by linear theory should time V max /6 (V max is the maximum of mean longshore current velocity), which will better reflect the disturbance magnitude in the experiment. Fig. 9 gives the superposed flow field (u', V+v') of the disturbance velocity field corresponding to the dominant unstable mode and the mean longshore current for regular wave (T=1.0 s, H=2.52 cm). Meanwhile, the averaged velocity field measured by the ADV flow meters along the coast in the experiment is also shown for comparison in Fig. 9.
It can be seen from Fig. 9 that the superposed flow field calculated by the linear instability analysis is similar to the velocity field obtained by experimental analysis. The swing wavelength is about 8 m, which can be measured by averaging the measurement of the corresponding CCD ink images of given wave cases, as shown in Fig. 10. Table 2 gives the comparison of the predicted wavelength calculated by linear instability analysis and the results displayed by ink in the experiment. The table shows only the regular wave cases which can determine clearly the wavelength from the CCD ink images and the ink motion pictures. For the irregular wave cases, the results are not given here be-   cause the ink is too discrete to be recognized clearly. Table 2 shows that the unstable wavelength calculated by the linear instability theory is close to the experimental results and the relative error is smaller than 15%. It indicates that the swing characteristics of superposed flow field calculated by the linear instability theory is similar to that observed in the experiment. Generally speaking, the linear instability theory can be used to explain the longshore current instability to a certain extent. It should be noted that linear instability theory can be well applied to the early stage of longshore currents instability. When the instability develops to a certain extent (Shen et al., 2017), the linear instability theory no longer satisfies the assumption of the small perturbations u', v' and η', so the nonlinear instability theory needs to be included.

Conclusions
This paper gives the mathematical description of longshore current linear instability model with the bottom friction being incorporated. The instability characteristics and the superposed flow field are obtained by linear instability model based on the mean longshore current velocity profile fitted by experimental results of slopes 1:100 and 1:40. The calculated results are compared with the experimental results and the main conclusions are as follows.
(1) The longshore current instability only exists in backshear mode for slope 1:40 and may exist in frontshear and backshear modes for slope 1:100. The peaks of linear instability growth mode for slope 1:100 include three cases: (a) The dominant peak is induced by joint-action of frontshear and backshear, the smaller peak is induced by frontshear or backshear. (b) The only major peak is induced by joint-action of frontshear and backshear. (c) The only larger peak is induced by backshear.
(2) The influence of bottom friction to longshore current instability will decrease the corresponding unstable growth rate, but it does not change the dominant unstable growth mode and its corresponding unstable wave number and propagation velocity.
(3) The results calculated by the linear instability analysis are coincident with those by spectral analysis. Meanwhile, the wavelength and spatial variation obtained by linear instability analysis are similar to the experimental results.