# The Behavior of Nonlinear Tsunami Waves Running on the Shelf
## 5 A Solution to the Nonlinear Problem: Turbulent Horizontal Friction Dominates  
In this case, \(R_{L}>>R_{B}\), and Equation (8) takes the form of Equation (13). Considering that \(\xi=\mu X\), \(F_{r}/\mathrm{Sh}=\xi_{*}/\mu(H-r)\) and choosing \(\zeta_{*}=(H-r)\), we rewrite Equation (13) as follows:  
\[\frac{\partial Z}{\partial X}\ +\ Z\frac{\partial Z}{\partial\tau}\ =\ \frac{F_{L}}{2}\ \frac{ \partial^{2}Z}{\partial\tau^{2}}. \tag{21}\]  
This is the Burgers equation, which occurs in gas dynamics and nonlinear acoustics [35; 36; 40; 41]. It is sometimes written with a different sign for a nonlinear term:  
\[\frac{\partial Z}{\partial X}\ -\ Z\frac{\partial Z}{\partial\alpha}\ =\ \lambda\ \frac{ \partial^{2}Z}{\partial\alpha^{2}}\, \tag{22}\]  
where \(\alpha=-\tau\), \(\lambda=F_{L}/2\). Let us introduce the function \(\Psi\): when \(Z=\partial\Psi/\partial y\). Then, from Equation (22), it follows that  
\[\frac{\partial\Psi}{\partial X}\ -\ \frac{1}{2}\ \bigg{(}\frac{\partial\Psi}{ \partial\alpha}\bigg{)}^{2}=\lambda\ \frac{\partial^{2}\Psi}{\partial\alpha^{2}}. \tag{23}\]  
We also introduce the function \(\varphi\): \(\psi\) = 2\(\lambda\) ln\(\varphi\). Then, Equation (23) takes the following form:  
\[\frac{\partial\phi}{\partial X}\ =\ \lambda\ \frac{\partial^{2}\phi}{ \partial\alpha^{2}}. \tag{24}\]  
This is the classical linear heat equation well-known in mathematical physics [42]. It can also be used to find a solution to the nonlinear Equation (22). It has the following form [41]:  
\[Z(\tau,\ X)\ =\ \sqrt{\frac{\lambda}{\pi\ X}}\ F\bigg{(}\frac{\tau}{2\ \sqrt{\lambda\ \ X}}\bigg{)}, \tag{25}\]  
where the function \(F\) is  
\[F(m)\ =\ \frac{\left[1-exp(-p)\right]\ exp\big{(}-m^{2}\big{)}}{1-\left[1- exp(-p)\right]\ V(m)} \tag{26}\]  
and \(p\) is a parameter determined by the initial conditions. It is also designated here:  
\[m\ =\ \frac{\tau}{2\ \sqrt{\lambda\ \ X}}, \tag{27}\]  
\[V(m)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{m}exp\Big{(}-w^{2}\Big{)}dw=\ \frac{1}{2}[1+\ erf(m)] \tag{28}\]  
and \(erf\) (\(m\)) is the Gaussian error integral [42]  
\[erf(m)=\frac{2}{\sqrt{\pi}}\int_{0}^{m}exp\Big{(}-w^{2}\Big{)}dw. \tag{29}\]For example, consider a shelf with an average depth of \(H=70.5\) m and a height of roughness ridges \(r=0.5\) m. Then, \((H-r)=70\) m and \(c_{0}=26.19\) m/s. We also set the value \(A_{L}=3930\) m\({}^{2}\)/s. Such a large value of horizontal turbulent viscosity is typical for large-scale ocean circulation models [30]. Let a tsunami wave fall on the shelf, and we observe its arrival at the initial time \(t=0\) at the point \(x=100\) m from the sea edge of the shelf (Figure 4). We see that the leading edge twists as the wave amplitude and nonlinearity increase, and the wave tends to break. On the contrary, small amplitude waves are almost symmetrical and smoothed out due to the large turbulent viscosity. The nonlinear Equation (22) also has other solutions in the form of solitary waves (solitons). They arise due to the balance of nonlinearity and friction. Here is another example of a tsunami soliton. It is obtained by solving the auxiliary heat conduction (Equation (24)). It looks like  
\[\phi\ =\ A\ +\ \mathrm{erf}\ [\alpha/(4\ \lambda\ X)], \tag{30}\]  
where \(A\) is the integration constant determined by the initial conditions. Let us substitute Equation (30) into the relationship between the functions \(\psi\) and \(\varphi\), then into the definition of the value of \(Z\) through the function \(\psi\). As a result, we obtain the desired solution of Equation (22):  
\[\frac{\partial Z}{\partial X}\ -\ Z\frac{\partial Z}{\partial\alpha}\ =\ \lambda\ \frac{ \partial^{2}Z}{\partial\alpha^{2}}\, \tag{22}\]0  
Figure 5 displays Equation (31)'s calculation of three tsunami solitons corresponding to different values of the constant \(A\): blue curve at \(A=3\), red curve at \(A=4\), and green curve at \(A=7\).