⇦Chinese Physics Letters, 2019, Vol. 36, No. 9, Article code 090501 Formation of Square-Shaped Waves in the Biscay Bay * Xin Li (李欣)1,2, Wen-Hao Xu (许文昊)1,2, Dong-Ming Chen (陈栋铭)1,2, Li-Ke Cao (曹利克)1,2, Zhan-Ying Yang (杨战营)1,2** Affiliations 1School of Physics, Northwest University, Xi'an 710069 2Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069 Received 26 April 2019, online 23 August 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11875220 and 11425522.
**Corresponding author. Email: zyyang@nwu.edu.cn Citation Text: Li X, Xu W H, Chen D M, Cao L K and Yang Z Y et al 2019 Chin. Phys. Lett. 36 090501    Abstract Recently, a report from Elite Readers suggested that a strange phenomenon of 'square-shaped waves' had occurred at the beaches of the Isle of Rhe in the Bay of Biscay. Based on the hydrological and geological data of the Bay of Biscay, we find that the special phenomenon is closely related to a solitary wave that can be described by the shallow water wave equation. We discuss the formation mechanisms of the square-shaped waves by the Kadomtsev–Petviashvili equation. The combination of exact solutions and actual condition provides the simulated initial state. We then reproduce a square-shaped structure by a numerical method and obtain the result consistent with the observed picture from media. Our work enriches public understanding of strange water waves and has great significance for tourism development and shipping transportation. DOI:10.1088/0256-307X/36/9/090501 PACS:05.45.Yv, 92.10.Hm, 47.35.-i © 2019 Chinese Physics Society Article Text Water waves are universal and complex phenomena affected by the geographical factors in nature. The study of water waves has a long research history and continuous improvement, while the science has made constant progress. The relative theories developed from original tidal theory, linear water wave theory to nonlinear water wave theory. One of the essential theories in the depiction of water waves is the solitary wave theory. Russell reported the first observational record of a solitary wave in 1834.[1] The solitary wave propagated with a constant velocity and shape for one or two miles along a canal. Widespread attention has been taken to this particular phenomenon. However, it was not reflected well by scientific explanation at that time.[2,3] After sixty years of unremitting efforts, Korteweg and Vries derived approximate nonlinear equations for shallow water waves and presented the accurate mathematical expression in 1895.[4,5] Zabusky and Kruskal found that the solitary waves of the KdV equation have remarkable elastic interaction properties and called them 'solitons'.[6] Their work is one of the notable results in the progress of water wave study,[7] which has been evaluated as a milestone in the PRL's 50th anniversary and opens the gates to soliton research in other fields such as nonlinear optics,[8] plasma,[6] neurography[9] and finance.[10] Nowadays, there is a general consensus on existence of rogue waves in oceans.[11] One remarkable feature of rogue waves is that they appear visibly from nowhere and disappear without a trace. Solitary wave theory is one of the approaches that has been successful in predicting the basic features of rogue waves.[12–15] One of the prototypes suggested to model of rogue waves is the so-called Peregrine soliton.[16,17] In a manner of speaking, the observation and study for rogue waves have stimulated a series of new researches about nonlinear optical rogue waves physics,[18–20] magnetic rogue waves physics[21] and matter rogue waves physics.[22] Nevertheless, there are plenty of strange and unexplained phenomena in nature.
cpl-36-9-090501-fig1.png
Fig. 1. Square-shaped waves in the Isle of Rhe. Photos are taken from www.elitereaders.com/rare-dangerous-square-shaped-waves.
A newspaper in France recently reported that a peculiar water wave structure appeared at the beaches of the Isle of Rhe in the Bay of Biscay (46.20$^{\circ}$N, 1.40$^{\circ}$W),[23] as shown in Fig. 1. The Bay of Biscay is one of the most internationally famous zones for heavy storms and waves. However, the wave structure near the isle shows stable square shape, which has been called 'square-shaped waves' (also called 'grid waves') in news reports. What kind of water wave is this? Can we depict this phenomenon in physical models? And, why does the phenomenon only appear on the beach of Rhe? By analyzing the geographical data of the Bay of Biscay, it is found that the square structure wave is a kind of shallow water wave. Thus we apply the Kadomtsev–Petviashvili (KP) equation to investigate the formation mechanism and create a model of square-shaped waves.[2] The shape of water wave in surface can be described by solution of the KP equation.[24–27] However, wind disturbance and the interaction between the North Atlantic currents may result in water waves from any direction, only different wave trains traveling at opposite angles (positive and negative angles with respect to $x$ direction) can form the square-shaped waves. Consequently, we set the corresponding initial state and use an integral factor scheme for simulation.[28,29] The evolution results conform to the figures of the observation on square-shaped waves. The Isle of Rhe is located on the eastern part of the Bay of Biscay, which lies in east of North Atlantic between the France and Spain, as shown in Fig. 2(a). The Bay of Biscay is a part of the route from Gibraltar to Dover, hence the bay possesses crucial transportation function and military strategic significance. The bay is also known for its massive internal tides.[30] Azevedo et al. investigated available satellite image and revealed that the Bay of Biscay is a hotspot region and has a high level of internal tide activity in 2006.[31] The estuary of the Bay of Biscay is towards the west, which is characterized by a shape that is narrow and deep inside, and wide and shallow outside. Consequently, the area has a trumpet-shaped structure. There are many other trumpet-shaped structures across the world, such as Qiantang River in China, the Fundy bay in Canada, Seine estuary in France and the Thames Estuary in England.[32–34]
cpl-36-9-090501-fig2.png
Fig. 2. (a) The geographical location of the Bay of Biscay. (b) The contour line near the Isle of Rhe.[35]
However, the special phenomenon of square-shaped waves, has never been reported in these places and can only be observed at the beach near the island. Thus, we plot the contoured bathymetric chart of the Bay of Biscay with the related geographical data from NNAC.[35] As shown in Fig. 2(b), the water depth distribution is varied near the island: it decreases gradually from more than 4 km in the west to only a few meters in east. Near this beach, the depth is only tens of meters. Thus, this strange phenomenon can be classified as a shallow water wave in an ocean. As a starting point of our study, we address the problem via the shallow water equation; i.e., the KP equation. The KP equation is a nonlinear partial differential equation in two-spatial and one-temporal coordinates, which describes the slow evolution of nonlinear and long waves of small amplitudes on transverse coordinates.[2,24] In recent years, it has been widely used in studying ocean water waves and tides in bays or gulfs.[36–38] There is a general expression of the KP equation, which can be written in normalized form as follows:[39] $$\begin{alignat}{1} \Big(\frac{1}{c_{0}}{\eta}_{t}+{\eta}_{x}+{\dfrac{h^2}{2}\sigma}{\eta}_{xxx} +{\dfrac{3}{2h}}{\eta}{\eta}_{x}\Big)_{x}+\frac{1}{2}{\eta}_{yy}=0,~~ \tag {1} \end{alignat} $$ where $\eta(x,y,t)$ is the tidal height, $x$ and $y$ are the longitudinal and transverse spatial coordinates, respectively, and $t$ is the time. The subscripts $x$, $y$, $t$ denote partial derivatives. The parameter $g$ is the gravitational acceleration, $h$ is the depth of water, and $c_{0}=\sqrt{g h}$ is the phase velocity. Moreover, $\sigma=(1/3-\hat{T})$ represents the surface tension and $\sigma \approx 1/3$ for the salinity data in the North Atlantic, and $\hat{T}=T/\rho gh$, where $T$ and $\rho$ represent the tension coefficient and density, respectively. By introducing the scaling transform: $t'=\beta t$, $\beta=2c_{0}\sigma/h$, $x'=(x-c_{0}t)/h$, $y'=y/h$, $\eta'=2\sigma hu$, and omitting the superscript, we can obtain the dimensionless equation $$\begin{align} (4{u}_{t}+{u}_{xxx}+6uu_{x})_{x}+3u_{yy}=0.~~ \tag {2} \end{align} $$ The soliton solutions to Eq. (2) are obtained through the transformation[40–42] $$\begin{align} u(x,y,t)=2\partial_{x}^{2}[{\ln}\tau(x,y,t)].~~ \tag {3} \end{align} $$ For a pre-selected set of wave numbers, $\{k_{1},\ldots, k_{r}\}$, the function $\tau(x,y,t)$ is given by $$\begin{align} \tau=|E\cdot A^{T}|,~~ \tag {4} \end{align} $$ where the details of matrices $E$ and $A$ are as follows: $$ E=\left| \begin{array}{ccccc} E_{1} & E_{2} &\ldots &\ldots& E_{r} \\ k_{1}E_{1} & k_{2}E_{2} &\ldots &\ldots& k_{r}E_{r}\\ \vdots & \vdots & \ddots & \ddots & \vdots\\ k_{1}^{l-1}E_{1} & k_{2}^{l-1}E_{2} &\ldots&\ldots& k_{r}^{l-1}E_{r}\\ \end{array}\right|, $$ $$ A=\left| \begin{array}{ccccc} a_{11} & a_{12} &\ldots &\ldots & a_{1r} \\ a_{21} & a_{22} &\ldots &\ldots & a_{2r}\\ \vdots & \vdots &\ddots & \ddots & \vdots\\ a_{l1} & a_{l2} &\ldots &\ldots & a_{\rm lr}\\ \end{array}\right|, $$ with $E_{m}=\exp(k_{m}x+k_{m}^{2}y-k_{m}^{3}t),~~m=1,2,3,\ldots,r$. Let us present the simplest form, one line-soliton solution, which corresponds to common water wave shapes. A line-soliton solution is obtained by a $\tau$ function with two exponential terms; i.e., the case $l=1$ and $r=2$ in Eq. (4). The form of $A$-matrix is $A=(1 a)$, then we have the line-soliton solution $$\begin{align} u(x,y,t)=\,&H{\rm{sech}}^{2}(K [ x-\psi\,y+c\,t]), \\ H=\,&\frac{1}{2}(k_{j}-k_{i})^{2},~~K=\frac{1}{2}(k_{j}-k_{i}), \\ \psi=\,&k_{i}+k_{j},~~c=k_{i}^{2}+k_{i}k_{j}+k_{j}^{2},~~ \tag {5} \end{align} $$ where $H$ is the amplitude of solitary wave, $\psi$ is the tangent value of the angle between soliton's propagation direction and the positive $x$-axis, and $c$ is the propagation velocity of water waves. In fact, the exact solution cannot describe the actual pictures of water wave in the Bay of Biscay comprehensively. However, it still provides a reference for us to study the initial state. We choose an initial condition, which is formed by gluing together pieces of line-soliton solutions. In particular, gluing means using half of line soliton and half of another line solitons with different $k_{i}$ and $k_{j}$ respectively and joining the ends together. The details of the initial state are given as follows: $$\begin{align} u_{0}=\,& \sum^{4}_{i=1}H_{i}{\rm{sech}}^{2}[\frac{1}{2}K_{i}(x+\psi_{i} (y+d_{i}))]\times \varepsilon(y)\\ &+\sum^{8}_{i=5}H_{i}{\rm{sech}}^{2}[\frac{1}{2}K_{i}(x+\psi_{i} (y+d_{i}))]\times \varepsilon(-y),~~ \tag {6} \end{align} $$ where $H_{i}$ represents the amplitude of each solitary wave, $\psi_{i}$ is the tangent value of the angle between soliton's propagation direction and the positive $x$-axis, $d_{i}$ is a constant which can change the space between lines, and $\varepsilon(y)$ is the Heaviside step function; i.e., $$ \varepsilon(y)=\left\{\begin{aligned} 1,& ~~y \ge 0, \\ 0,& ~~y < 0. \end{aligned}\right. $$ We present numerical simulations which show the evolution of various types of solitary wave initial data to the exact solution of the KP equation. We solve the KP equation numerically using the integral factor scheme on a domain $D=\{(x,y):-L_{x}/2 < x < L_{x}/2,-L_{y}/2 < y < L_{y}/2\}$. We assume that solution is periodic in both $x$ and $y$, and rescales the domain to a fixed domain[43] $D=\{(x,y):-\pi/2 < x < \pi,-\pi/2 < y < \pi\}$ by translation $X=(2\pi/L_{x})x$, $Y=(2\pi/L_{y} )y$. Thus Eq. (2) becomes $$\begin{align} &({u}_{t}+Q{u}_{xxx}+P{u}{u}_{x})_{x}+R{u}_{yy}=0, \\ &P=\dfrac{3\pi}{L_{x}},~Q=\dfrac{2\pi^{3}}{L_{x}^3},~~R=\dfrac{3 L_{x}}{2L_{y}^2}.~~ \tag {7} \end{align} $$ We can express the solution by fast Fourier transform operation due to periodicity. The operation is $\hat{u}{(l,m,t)}=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} u(X,Y,t)e^{-i(lX+mY)}dl\,dm$. Then, the solution reduces into a differential equation for the time evolution of $\hat{u}{(l,m,t)}$, i.e., $$\begin{alignat}{1} \hat{u}_{t}+i\dfrac{lP}{2}F(\hat{u})+i\Big(\dfrac{Rm^2}{l}-Ql^{3}\Big)\hat{u}=0,~~ l\neq0,~~ \tag {8} \end{alignat} $$ where $F(\hat{u})$ is the Fourier transform of $u^{2}$, which is numerically evaluated as $F(\hat{u})={\rm FFT}[({\rm IFFT}[\hat{u}])^2]$ with ${\rm FFT}$ and ${\rm IFFT}$ the fast Fourier transform operation function and the inverse fast Fourier transform operation function which can be calculated in Matlab directly. For convenience, we introduce transformations: $c=i(Rm^2/l-Ql^3)$, $\hat{u}=\hat{u}e^{-ct}$, and $d=ilP/2$. Hence Eq. (8) can be expressed as follows: $$\begin{align} \hat{u}_{t}+de^{\rm ct}F(\hat{u}e^{-ct})=0.~~ \tag {9} \end{align} $$ The simplified form (Eq. (9)) is a standard differential equation which can be numerically solved using the 4-order Runge–Kutta method for a given initial data $\hat{u}(l,m,0)$ obtained from the Fourier transform of $u(x,y,0)$.[30] The solution $u(X,Y,t)$ and $u(x,y,t)$ is then reconstructed by taking the inverse Fourier transform of $\hat{u} (l,m,0)$ for $l\neq0$.
cpl-36-9-090501-fig3.png
Fig. 3. (a) The random initial state. (b) The evolution result in $t=10$. (c) Actual picture of ocean surface.[44] The corresponding parameters are set as: $H_{i}=0.5$, $K_{i}=1$, $\{\psi_{1},\psi_{2},\psi_{3},\psi_{4}\}=\{-1.6,-1.8,-2.04,-2.24\}$, $\{\psi_{5},\psi_{6},\psi_{7},\psi_{8}\}=\{1.6,1.8,1.7,1.9\}$, $\{d_{1},d_{2},d_{3},d_{4}\}=\{0,7,12,15\}$, $\{d_{5},d_{6},d_{7},d_{8}\}=\{0,-6,-11,-16\}$. (d) The initial state of special evolution. (e) The evolution result in $t=10$. (f) Actual picture of ocean surface.[24] The corresponding parameters are set as $H_{i}=0.5$, $K_{i}=1$, $\psi_{1\leq i\leq10}=-1.4$, $\psi_{11\leq i\leq20}=1.4$, {$d_{i}(1\leq i \leq10)\}=0,5,10,15,20,25,30,35,40,45$, $\{d_{i}(11\leq i \leq20)\}=\{0,-5,-10,-15,-20,-25,-30,-35,-40,-45\}$.
Let us consider the general case of water waves. The Bay of Biscay is located in the prevailing westerly zone and coincides with the mean position of the polar front. The frequent activities of cyclones often result in high winds and waves. Under the influence of the North Atlantic circulation, the currents in the bay also move clockwise. However, because of the wind disturbance and the interactions among the North Atlantic currents, North Pacific currents and the Kuroshio, the original currents may be disturbed and this can result in water waves arriving from any direction. We start the generation of grid waves by the initial states of Eq. (6). For the sake of simplicity, we set amplitude of common initial states $H_{i}=0.5$ and set the direction and interval between wave trains of initial state as random values. We then generate a strange initial state of water waves on the surface, as shown in Fig. 3(a). Using the integral factor scheme, we can observe the different time stages of wave train evolution. Figure 3(b) shows the irregular shapes in the propagation at $t=10$. Under the impacts of the wind's disturbance and the interaction between different currents, the initial currents may be disturbed and flow towards many directions. The structure will form irregular grid waves as time goes on. Figure 3(c) shows an actual wave similar to the numerical result. We should reinforce the point that our model mainly describes the evolution of water wave's structure based on the classical solitary theory, and just establish qualitative relationship between the model's parameters and environment factors of climate. In particular, the North Atlantic currents passes the Bay of Biscay during each summer. The currents may generate different wave trains that travel at opposite angles because of the trumpet-shaped structure and the prevailing westerly wind. The square-shaped waves on the surface are formed when the two wave trains travel for a long period of time. Thus, we choose the directions of two wave trains as opposite angles $\psi_{1\leq i\leq10}=-1.4$, $\psi_{11\leq i\leq20}=1.4$. The initial state of square-shaped waves is shown in Fig. 3(d). The square-shaped waves hold the cross grid shape in the propagation (see Fig. 3(e)), which coincides with the actual situation as shown in Fig. 3(f). If the initial parameters of wave trains (such as the amplitude, velocity and interval) are changed, then the amplitude, period and shape of square-shaped waves will be changed but they can still be generated and remain stable. In addition to $t=10$, at some other times, one can also observe that the formation and holding of square-shaped waves. The square-shaped waves can keep their features for a long time ($t=35$). Due to cumulative error, the simulation cannot describe the results very well with the continuing increase in time. Consequently, the numerical simulations verify this assumption, which means that the strange wave dynamics can be described by the KP equation, and square-shaped waves can be formed by superimposing two chains of solitons. Similarly, many structures can be obtained by our theoretical model under the reasonable settings. The $N$-soliton of nonlinear dynamics may be able to provide a new path to investigate the complicated water waves in nature. It also helps to understand the formation mechanism of square-shaped waves and irregular grid waves. In summary, we have discussed the formation mechanism of square-shaped waves, which is reported by media in recent years based on the known data. We take the KP equation to study the phenomenon with the aid of numerical technology. We assume that the initial state is made up of exact solutions and obtain a numerical result that is consistent with the observation data. Square-shaped waves can be generated by such regular wave trains and their features can be controlled by the initial parameters. These waves can remain stable for a long time. Similarly, for irregular initial wave trains, irregular grid waves can also form, as a more universal phenomenon. However, several other causes that form square-shaped waves cannot be discussed in this work due to the lacking of hydrological data and meteorological data about the Isle of Rhe in the Bay of Biscay. We expect that our results will provide some references for further research about square-shaped waves. It may also have a considerable influence on transportation and the tourism development of the Isle of Rhe. References Solitary WavesXLI. 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