# The Behavior of Nonlinear Tsunami Waves Running on the Shelf
## 2 Formulation of the Problem  
We chose the coordinates' origin at the shelf's sea edge \(x=0\). We directed the \(x\)-axis along the direction of wave propagation perpendicular to the coast, the \(y\)-axis as perpendicular to the \(x\)-axis (left), and the \(z\)-axis as vertically down (Figures 1 and 2). The letter \(M\) denotes the width of the shelf. The level \(z=0\) is located on the surface of calm water; the letter \(z\) denotes the wave disturbance of the sea surface; the positive value \(z\) is counted down from the undisturbed level \(z=0\) (Figure 2). The letter \(H\) denotes the average shelf depth (effects of variable depth and wave refraction are not considered here), and the letter \(r\) denotes the average height of the rough protrusions on the bottom so that the total depth of the shelf is \(H-r\). We used the shallow water theory equations. They are obtained from geophysical hydrodynamic equations by integrating along the \(z\)-axis in the range from \(z=\zeta\) to \(z=(H-r)\)[4; 31; 32]. Assuming that there are no changes along the \(y\)-axis (\(\partial/=0\)), we write these equations in the following form:  
\[\frac{\partial\xi}{\partial t}+\frac{\partial S}{\partial x}\ =0, \tag{1}\]  
\[\frac{\partial S}{\partial t}+n\frac{\partial S^{2}}{\partial x}\ =\ c_{0}^{2}\frac{ \partial\xi}{\partial x}+A_{L}\frac{\partial^{2}S}{\partial x^{2}}-R_{x}^{H}. \tag{2}\]Here, \(\zeta\) is the perturbation of the water surface level, \(n=1/(H-r)\), \({c_{0}}^{2}=g\left(H-r\right)\), \(A_{L}\) is the coefficient of horizontal turbulent viscosity, and \(g\) is the gravity acceleration. The turbulent friction on the bottom \({R_{x}}^{H}\) can be related to the total flow rate \(S\) by a linear law:  
\[R_{x}^{H}=\omega_{T}S,\ \omega_{T}=\frac{3A}{\left(H-r\right)^{2}},S=\int_{\zeta}^{H -r}udz. \tag{3}\]  
Here, \(\omega_{T}\) is the relaxation frequency (friction), \(A\) is the coefficient of vertical turbulent viscosity [33], and \(u\) is the flow velocity component in the wave along the \(x\)-axis. We also consider another nonlinear friction law below at the end of this article. Equation (2) can be conveniently reduced to the dimensionless form:  
\[\frac{\partial^{2}\Pi}{\partial T^{2}}\ +Fr\frac{\partial^{2}\Pi^{2}}{ \partial T\partial X}+F_{B}\frac{\partial\Pi}{\partial T}\ -F_{L}\frac{\partial^{3}\Pi}{\partial T\partial X^{2}}-\frac{ \partial^{2}\Pi}{\partial X^{2}}\ =\ 0. \tag{4}\]  
Here, \(T=\omega t\), \(\Pi=S/S\), \(X=kx\), the relationship between frequency \(\omega\) and wavenumber \(k\) is given by \(\omega=c_{0}k\), \(Fr=u_{*}/c_{0}\) is the Froude number, \(F_{B}=\omega_{T}/\omega\) is the bottom friction, \(F_{L}=Fr/\text{Re}=\omega A_{L}/c_{0}^{2}\) is the horizontal friction parameter, \(\text{Re}=(u_{*}/k)A_{L}\) is the Reynolds number, and \(S*=u_{*}/n=u_{*}\left(H-r\right)\) is the total flow rate. The letters with asterisks denote the characteristic values of the corresponding quantities. For example, \(u*\) is the characteristic flow velocity in the wave. For a tsunami wave on the shelf, we can take \(\omega=1745\ \times 10^{-3}\) Hz, \((H-r)=70\) m, \(c_{0}=26.20\) m/s, \(A=0.0285\) m\({}^{2}\)/s, \(A_{L}=3.93\cdot 10^{3}\) m\({}^{2}\)/s, and \(u_{*}=0.262\) m/s. Then, \(Fr=F_{B}=F_{L}=\mu\), where \(\mu\) is a small parameter (\(\mu=0.01\)). Figure 1: Simplified scheme: horizontal coordinate axes and symbols (modified after [1]). If \(\mu=0\), then Equation (4) transforms into a wave equation with a solution:  
\[\Pi=\Theta\left(T-X\right)=\Theta\left(\omega t-kx\right). \tag{5}\]  
Here, the function \(\Theta\) is determined by the initial conditions. Therefore, when \(\mu\
eq 0\), it is natural to look for the solution of Equation (4) in the following form:  
\[\Pi=\Lambda\left(\mu X;T-X\right), \tag{6}\]  
introducing slow \(\xi=\mu\)\(X\) and fast \(\tau=\left(T-X\right)\) variables.