2.1 One-dimensional (horizontal) linear and fully dispersive model

2.2 One-dimensional (horizontal) linear and weakly dispersive model

The linear and weakly dispersive model (LWD) can be obtained by linearising the Boussinesq equations (Nwogu Reference Nwogu1993) or the fully nonlinear and weakly dispersive model (Wei et al. Reference Wei, Kirby, Grilli and Subramanya1995; Lynett & Liu Reference Lynett and Liu2002). Alternatively, it can be obtained as an approximated version of the LFD based on the long-wave assumption (see, for example, Lynett & Liu (Reference Lynett and Liu2002), who demonstrated how to obtain weakly dispersive models from a fully dispersive model). The LWD is accurate when both $O(\unicode[STIX]{x1D716})\ll 1$ and $O(\unicode[STIX]{x1D707}^{4})\ll 1$ . In weakly dispersive models, the governing equations are often solved at a representative depth $z_{\unicode[STIX]{x1D6FC}}$ , which is chosen to improve the characteristics of frequency dispersion; the horizontal velocity at depth $z=z_{\unicode[STIX]{x1D6FC}}$ is written as $u(x,z_{\unicode[STIX]{x1D6FC}},t)=u_{\unicode[STIX]{x1D6FC}}(x,t)$ . For the LWD in constant depth, $z_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D6FC}$ since the normalised water depth is $1$ , and here we use $\unicode[STIX]{x1D6FC}=-0.531$ as recommended by Nwogu (Reference Nwogu1993). The dimensionless continuity equation reads

and the momentum equation reads

where $\unicode[STIX]{x1D6FC}_{11}$ , $\unicode[STIX]{x1D6FC}_{12}$ , $\unicode[STIX]{x1D6FC}_{21}$ and $\unicode[STIX]{x1D6FC}_{22}$ are constants:

Solving for $u_{\unicode[STIX]{x1D6FC}}$ first is more convenient in this case, and (2.14) and (2.15) can be combined to yield

The LWD can be solved in a manner similar to those for the LSWEs and the LFD and here we only outline the solution process. When $u_{\unicode[STIX]{x1D6FC}}$ is solved for first, the imposed initial conditions are

The first condition results from setting $u(x,0,0)=0$ (initial horizontal velocity on the free surface is zero) and recovering $u_{\unicode[STIX]{x1D6FC}}(x,0)$ from the velocity profile assumed in the LWD. The second condition results from setting $\unicode[STIX]{x1D702}_{x}(x,0)=0$ in (2.15); thus, $\unicode[STIX]{x1D702}(x,0)=\text{const}$ . If the derivatives of $u_{\unicode[STIX]{x1D6FC}}$ and $B$ are all continuous, the order of differentiation does not matter. By Laplace–Fourier-transforming (2.17) and applying the above initial conditions, equation (2.17) becomes

from which $\overline{\widetilde{u_{\unicode[STIX]{x1D6FC}}}}$ can be solved for as

The $\overline{\widetilde{\unicode[STIX]{x1D702}}}$ can then be recovered from transformed (2.14) or (2.15). If (2.14) is used, we obtain

Since $u_{\unicode[STIX]{x1D6FC}}$ is solved for first in this case, the precise initial condition for the free surface, $\unicode[STIX]{x1D702}(x,0)$ , can be specified only when the solution for $\unicode[STIX]{x1D702}$ is considered. Here, we require $\unicode[STIX]{x1D702}(x,0)=0$ .

For a translating landslide moving at a constant speed $Fr$ so that $B(x,t)=B(x-Fr\,t)$ , a closed-form inverse Laplace transform of (2.21) is available, and we find $\unicode[STIX]{x1D702}(x,t)$ to be

where

is the normalised wave speed of the LWD. We again see the three-wave structure and frequency-dispersive effects. Equations (2.22) and (2.23) are highly similar to (2.6) and (2.7) as the LWD is essentially an expanded (approximated) version of the LFD (the verification is shown in appendix C), based on the long-wave assumption. The same asymptotic approximation method can be used to obtain the far-field leading-wave solutions of (2.22), and the results will be exactly the same as those for the LFD, since the leading-wave solution depends on only the leading-order $O(\unicode[STIX]{x1D707}^{2})$ frequency-dispersive effects which are included in the LWD. The leading wave in the resonance case ( $Fr=1$ ) therefore also grows at the rate $t^{1/3}$ .

Similarly, a closed-form inverse Laplace transform of (2.20) is available, and we can express $u_{\unicode[STIX]{x1D6FC}}$ , the horizontal velocity at depth $z_{\unicode[STIX]{x1D6FC}}$ , as

The full horizontal and vertical velocities ( $u$ and $w$ respectively) in the LWD can be expressed as

It should be noted that the expressions in (2.25) are not unique, since alternate $O(\unicode[STIX]{x1D707}^{2})$ terms may be used. For more details on recovering the velocity field from $u_{\unicode[STIX]{x1D6FC}}$ in weakly dispersive models, see, for example, Lynett & Liu (Reference Lynett and Liu2002).

2.3 Comparison of the analytical solutions

The analytical solutions based on different wave models (LSWE, LWD and LFD) are compared (integral-form solutions are numerically integrated when necessary) in this section for a selected example. Since LSWE has a theoretical truncation error of $O(\unicode[STIX]{x1D707}^{2})$ , LWD has a theoretical truncation error of $O(\unicode[STIX]{x1D707}^{4})$ and LFD has no restriction on $\unicode[STIX]{x1D707}$ , we expect the three models to yield essentially identical results for a very long landslide with small $\unicode[STIX]{x1D707}$ (recall that $\unicode[STIX]{x1D707}$ is water depth divided by landslide length). As the landslide length shortens, LSWE and LWD become less and less accurate.

We have found $\unicode[STIX]{x1D707}=0.2$ to be an intermediate landslide length for which LSWE is inaccurate but LWD remains accurate. For a Gaussian-shaped landslide with $\unicode[STIX]{x1D707}=0.2$ moving at speed $Fr=0.5$ , the free-surface elevation solutions are compared in figure 7 and the velocity solutions are compared in figures 8 and 9. While LSWE is clearly inaccurate for this problem set-up, the LWD solutions overall agree well with the LFD solutions.

A case with a very long landslide, $\unicode[STIX]{x1D707}=0.05$ , in which all three models are accurate, as well as a case with a shorter landslide, $\unicode[STIX]{x1D707}=0.4$ , in which neither LSWE nor LWD is accurate, are shown in appendix D.

Figure 7.

The free-surface elevation $\unicode[STIX]{x1D702}(x,t)$ at different times and locations, due to a moderately long Gaussian-shaped landslide with $\unicode[STIX]{x1D707}=0.2$ moving at speed $Fr=0.5$ : (a,d) $t=0.5$ ; (b,e) $t=1.0$ ; (c,f) $t=2.0$ ; solid line, LFD solution (2.6); dashed line, LWD solution (2.22); dash–dot line, LSWE solution (1.3); dotted lines, locations at which the velocity solutions are shown.

Figure 8.

The vertical distribution of the horizontal velocity $u(x,z,t)$ at different times and locations, due to a moderately long Gaussian-shaped landslide with $\unicode[STIX]{x1D707}=0.2$ moving at speed $Fr=0.5$ : (a,d,g) $t=0.5$ ; (b,e,h) $t=1.0$ ; (c,f,i) $t=2.0$ ; (a–c) $x=-1$ ; (d–f) $x=0$ ; (g–i) $x=1$ ; solid line, LFD solution (2.8); dashed line, LWD solution recovered from (2.25); dash–dot line, LSWE solution (1.4). It should be noted that the scale of the horizontal axis varies greatly from panel to panel.

Figure 9.

The vertical distribution of the vertical velocity $w(x,z,t)$ at different times and locations, due to a moderately long Gaussian-shaped landslide with $\unicode[STIX]{x1D707}=0.2$ moving at speed $Fr=0.5$ : (a,d,g) $t=0.5$ ; (b,e,h) $t=1.0$ ; (c,f,i) $t=2.0$ ; (a–c) $x=-1$ ; (d–f) $x=0$ ; (g–i) $x=1$ ; solid line, LFD solution (2.9); dashed line, LWD solution recovered from (2.25); dash–dot line, LSWE solution (zero vertical velocity). It should be noted that the scale of the horizontal axis varies greatly from panel to panel.

3.1 One-dimensional (horizontal) LSWEs

The dimensionless 1DH LSWEs read

and

where ( $\unicode[STIX]{x1D702}$ , $B$ ) are normalised by the landslide thickness $A$ , $x$ is normalised by the landslide length $L$ , $h=x\tan \unicode[STIX]{x1D703}$ , normalised by $L\tan \unicode[STIX]{x1D703}$ , is the still water depth on a constant slope of angle $\unicode[STIX]{x1D703}$ , $u$ is normalised by $A\sqrt{gL\tan \unicode[STIX]{x1D703}}/L\tan \unicode[STIX]{x1D703}$ and $t$ is normalised by $\sqrt{L/g\tan \unicode[STIX]{x1D703}}$ . The landslide forcing function $B$ is described in the water coordinate system (vertical–horizontal). The shoreline is located at $x=0$ and the physical domain of interest is $x>0$ .

Again, either $\unicode[STIX]{x1D702}$ or $u$ can be solved for first. To solve for $\unicode[STIX]{x1D702}$ first, equations (3.1) and (3.2) can be combined into

Tuck & Hwang (Reference Tuck and Hwang1972) first solved this equation using the Laplace and Hankel transforms to obtain the solution for $\unicode[STIX]{x1D702}$ . Here, we impose the initial conditions

the latter resulting from setting $[u(x,0)h]_{x}=0$ in (3.1) – on a constant slope with $h=x\tan \unicode[STIX]{x1D703}$ , this corresponds to $u(x,0)=\text{const.}/x$ – and rederive the integral-form solution (valid for $x>0,t>0$ ) as

where $\text{J}_{0}$ is the Bessel function of the first kind with order $0$ . The solution for $u$ can then be recovered from (3.2) as

The precise initial condition for the horizontal velocity, $u(x,0)$ , can be specified only when the solution for $u$ is considered; here, we require $u(x,0)=0$ for initially quiescent water.

While the solution for $\unicode[STIX]{x1D702}$ , equation (3.5), is in a slightly different form from that in Tuck & Hwang (Reference Tuck and Hwang1972), we have verified that the two solutions are indeed the same. Our goal is to seek special landslide forcing functions $B(x,t)$ that have closed-form integrals in (3.5). Liu et al. (Reference Liu, Lynett and Synolakis2003) showed that the use of a deforming Gaussian-like landslide moving at the same speed as the wave celerity reduces the number of integrals to one.

We consider a ramp-like landslide forcing function:

where $\unicode[STIX]{x1D6FD}$ is a constant to be chosen and $R$ is the ramp function, defined as

Thus, the temporal derivative of $B$ is

where $H$ denotes the Heaviside step function. A closed-form Hankel transform is available for the step function (Poularikas Reference Poularikas2000), and equation (3.5) can be written as

Furthermore, if we choose $\unicode[STIX]{x1D6FD}=1/2$ , the $\unicode[STIX]{x1D70F}$ -integrals can be evaluated (Rosenheinrich Reference Rosenheinrich2015), and the equation becomes

This last integral can also be evaluated (Gradshteyn & Ryzhik Reference Gradshteyn and Ryzhik2007), and we have simply

which is a jump from $t/6$ to $0$ travelling along $x=t^{2}/4$ , where the front of the landslide is. Similarly to the landslide used in Liu et al. (Reference Liu, Lynett and Synolakis2003), the landslide accelerates so that it always travels at local wave speed. The solution for $u$ can be recovered from (3.2), with the initial condition $u(x,0)=0$ , as

which is a step function whose amplitude does not change.

The analytical solution (3.12) serves as a challenging benchmark for numerical models, since it consists of a shock and a ‘shoreline’ ( $x\rightarrow 0$ ) in linear theory. To demonstrate this, we solved the dimensional 1DH LSWEs (3.1) and (3.2) using a finite-difference solver, which employs the fourth-order central difference scheme in space and the third-order strong stability preserving Runge–Kutta scheme in time (Gottlieb, Shu & Tadmor Reference Gottlieb, Shu and Tadmor2001). The open boundary condition (derivatives equal zero) is used on the right boundary. For the left (shoreline) boundary, linear extrapolation is used to acquire the surface elevations at the ghost points located at $x\leqslant 0$ , and the flow velocity at $x\leqslant 0$ is assumed to be zero, which are needed in computing the derivatives based on the central difference scheme. In addition, since the water depth at exactly $x=0$ is $0$ , the first point in the numerical model is actually located at $x=\unicode[STIX]{x0394}x$ , where $\unicode[STIX]{x0394}x$ is the grid spacing (normalised by the landslide length). Using two different grid spacings $\unicode[STIX]{x0394}x=0.01$ and $\unicode[STIX]{x0394}x=0.001$ , we computed the numerical solution (normalised) due to the ramp-like landslide forcing function, and the comparison with the analytical solution is shown in figure 10. While the general location of the jump as well as the water surface elevations on the two sides of the jump are well captured, the width of the jump greatly depends on the grid spacing $\unicode[STIX]{x0394}x$ . Overshoot of the computed surface elevation is obvious near the jump, which is a common characteristic of the finite-difference scheme near sharp gradients. On the other hand, the numerical shoreline treatment seems to perform well in this problem, although small oscillations can still be observed in the free-surface elevations away from the jump.

Figure 10.

Comparison of the numerical and analytical solutions of the 1DH LSWEs for a ramp-like landslide-generated wave on a constant slope at times $t=0.4,0.6,0.8,1$ ; solid line, numerical solution; dashed line, analytical solution; (a,c) numerical solution with grid spacing $\unicode[STIX]{x0394}x=0.01$ ; (b,d) numerical solution with grid spacing $\unicode[STIX]{x0394}x=0.001$ .

Even though the results presented in figure 10 appear to be satisfactory, it should be noted that the two grid spacings used in the example ( $\unicode[STIX]{x0394}x=0.01$ and $\unicode[STIX]{x0394}x=0.001$ ) are fairly fine and are impractical in larger simulation domains. Therefore, more sophisticated numerical methods are necessary to resolve the sharp gradient (jump) and the shoreline boundary with a more reasonably sized grid spacing in a numerical model based on linear theory, and the closed-form analytical solution (3.12) serves as a convenient tool to test the effectiveness of the numerical methods.

3.2 Two-dimensional (horizontal) LSWEs

The dimensionless 2DH LSWEs read

where the notations are the same as those in (3.1) and (3.2), with the addition of $v(x,y,t)$ , the horizontal water velocity in the $y$ -direction. On a plane beach of angle $\unicode[STIX]{x1D703}$ with the shoreline located at $x=0$ so that $y$ is in the longshore direction, the physical domain of interest is $x>0$ and the still water depth is $h(x,y)=x\tan \unicode[STIX]{x1D703}$ . Combining the equations to solve for $\unicode[STIX]{x1D702}$ first, we obtain

We seek the analytical solution to (3.16). It should be noted that, like in the 1DH case, the landslide forcing function $B(x,y,t)$ is described in the water coordinate system (vertical–horizontal).

The analytical solution to (3.16) for the landslide-wave problem, valid for $x>0,t>0$ , was first obtained in Sammarco & Renzi (Reference Sammarco and Renzi2008) and Renzi (Reference Renzi2010) with the initial conditions $\unicode[STIX]{x1D702}(x,y,0)=0$ and $\unicode[STIX]{x1D702}_{t}(x,y,0)=0$ , for any landslide forcing function symmetric about the $x$ -axis. However, we note that the initial condition $\unicode[STIX]{x1D702}_{t}(x,y,0)=0$ does not fulfil the quiescent water condition in the LSWEs and would result in initial water velocities that depend on $B_{t}(x,y,0)$ (which can be non-zero): by setting $\unicode[STIX]{x1D702}_{t}(x,y,0)=0$ in (3.14), we obtain $[u(x,y,0)h]_{x}+[v(x,y,0)h]_{y}=B_{t}(x,y,0)$ ; under this condition, $u(x,y,0)=0=v(x,y,0)$ is not possible for $B_{t}(x,y,0)\neq 0$ .

Similarly to the 1DH cases discussed previously, when solving for $\unicode[STIX]{x1D702}$ first, to allow for the quiescent water condition, the initial conditions that should be imposed are

the latter resulting from setting $[u(x,y,0)h]_{x}+[v(x,y,0)h]_{y}=0$ in (3.14), which allows for the possibility that $u(x,y,0)=0=v(x,y,0)$ . The precise initial velocities, $u(x,y,0)$ and $v(x,y,0)$ , can be specified only when the solutions for $u(x,y,t)$ and $v(x,y,t)$ are considered. It should be noted that the two different initial conditions used, i.e. $\unicode[STIX]{x1D702}_{t}(x,y,0)=0$ and $\unicode[STIX]{x1D702}_{t}(x,y,0)=B_{t}(x,y,0)$ , become identical only when a landslide forcing function with $B_{t}(x,y,0)=0$ is chosen.

We rederive the analytical solution using the new initial conditions and obtain

where $L_{n}(x)$ is the $n$ th-order Laguerre polynomial, $\overline{B}$ is the Fourier-transformed $B$ with respect to $y$ ( $B$ is assumed to be symmetric about the $x$ -axis) and $\unicode[STIX]{x1D706}=k(2n+1)$ . The velocity solutions based on the new solution for $\unicode[STIX]{x1D702}$ , equation (3.18), can be recovered from the momentum equation (3.15) as

and

For initially quiescent water, we further require $u(x,y,0)=0=v(x,y,0)$ , in addition to the initial conditions already imposed when solving for $\unicode[STIX]{x1D702}$ , i.e. $\unicode[STIX]{x1D702}(x,y,0)=0$ and $[u(x,y,0)h]_{x}+[v(x,y,0)h]_{y}=0$ .

The analytical solutions in all existing studies (e.g. Sammarco & Renzi Reference Sammarco and Renzi2008; Renzi Reference Renzi2010; Renzi & Sammarco Reference Renzi and Sammarco2010, Reference Renzi and Sammarco2012; Seo & Liu Reference Seo and Liu2013) include an extra term in addition to (3.18) due to non-zero initial water velocities:

Therefore, in this study, we only need to focus on the term (3.21) to evaluate how it affects previous findings. Clearly, the extra term vanishes if a landslide forcing function with $B_{t}(x,y,0)=0$ is used.

To examine the significance of $\unicode[STIX]{x1D702}_{extra}$ , we follow the asymptotic analysis discussed in Sammarco & Renzi (Reference Sammarco and Renzi2008), Renzi (Reference Renzi2010) and Seo & Liu (Reference Seo and Liu2013), and find the large- $t$ asymptotic approximation of $\unicode[STIX]{x1D702}_{extra}$ for $y>0$ to be

where

Equation (3.22) is the extra term included in the asymptotic solution in previous studies. It has the same form as a part of the asymptotic approximation of the true solution (3.18), the key difference being the $C_{n}$ term. Similar terms exist in the asymptotic approximation of (3.18). These terms need to be numerically evaluated and may depend on time, but in general their amplitudes do not grow in time for the translating landslide forcing functions considered in both previous studies and the present study. Therefore, while the asymptotic decay rate $t^{-1/2}$ remains the same, the exact wave shape can be very different due to the difference in the initial conditions used.

Figure 11.

Wave fields generated by a double-Gaussian landslide forcing function moving at a constant speed $1$ from $x=0$ . The solution is symmetric about the $x$ -axis. (a,d,g) Numerical solution with $\unicode[STIX]{x0394}x=\unicode[STIX]{x0394}y=0.01$ ; (b,e,h) $50$ -mode analytical solution; (c,f,i) $50$ -mode $\unicode[STIX]{x1D702}_{extra}$ ; (a–c) $t=1$ ; (d–f) $t=2$ ; (g–i) $t=3$ .

In previous studies, translating landslides moving at a constant speed, for which $B_{t}(x,y,0)\neq 0$ , were used in plotting and further examining the solutions. Choosing as an example a double-Gaussian landslide forcing function moving at a constant speed $1$ from $x=0$ , i.e. $B(x,y,t)=\text{e}^{-8(x-t)^{2}}\text{e}^{-8y^{2}}$ , we solved the 2DH LSWEs numerically using a similar numerical model to that described in the previous section (with a fairly fine grid spacing of $\unicode[STIX]{x0394}x=\unicode[STIX]{x0394}y=0.01$ ), and compare the results with the $50$ -mode analytical solution (3.18) evaluated with numerical integration methods described by Seo & Liu (Reference Seo and Liu2013). The resulting free-surface elevations at $t=1,2,3$ are shown in figure 11. Although the numerical solution of the LSWEs can be problematic near the shoreline, it is robust and reliable away from the shoreline. Since the integral-form analytical solution requires further numerical evaluation and approximation in plotting, in this case, the numerical solution is a better choice as the reference solution when the overall wave field (i.e. not the edge waves) is of concern, such as those shown in figure 11. While the numerically evaluated analytical solutions generally agree well with the numerical solutions, some oscillations are clearly visible (e.g. along $y=2$ at $t=2$ ). On the other hand, we see that in this example the magnitude of $\unicode[STIX]{x1D702}_{extra}$ is always significant when compared with the true solution, suggesting that its impact on the true solution is not negligible.

Figure 12.

Analytical solutions of the free-surface elevation along $y=2$ due to a double-Gaussian landslide forcing function moving at a constant speed $1$ from $x=0$ : solid line, numerical solution with $\unicode[STIX]{x0394}x=\unicode[STIX]{x0394}y=0.01$ ; dashed line, $6$ -mode analytical solution; dash–dot line, $10$ -mode analytical solution; dotted line, $50$ -mode analytical solution; (a) $t=1$ ; (b) $t=2$ ; (c) $t=3$ .

Figure 13.

Plot examining the convergence rate of the analytical solution for $\unicode[STIX]{x1D702}(2.5,2,2)$ : solid line, analytical solution as a function of the number of modes considered; dashed line, numerical solution.

Upon closer inspection of the oscillations observed in the analytical solutions in figure 11, we found the cause to be a poor convergence rate of the analytical solution with respect to the number of modes considered in (3.18). To highlight this issue, we focus on the wave profile along $y=2$ in figure 11, which appears to be highly problematic, and compare in figure 12 the numerical solution with analytical solutions computed using different numbers of modes. We see that in this case, convergence of the solution is not reached even with $50$ modes considered. To further examine the convergence rate, in figure 13 we pick a point $(x,y,t)=(2.5,2,2)$ and plot $\unicode[STIX]{x1D702}(2.5,2,2)$ as a function of the number of modes considered. Based on figure 11, this point is situated ahead of the leading wave. We thus expect a very small value here – the numerical solution predicts $\unicode[STIX]{x1D702}(2.5,2,2)\simeq 0.008$ , yet the analytical solution is still only slowly converging even with 300 modes considered. We remark that the numerical integration can be time-consuming, and it is often undesirable to consider more than ${\sim}20$ modes. As a reference, Sammarco & Renzi (Reference Sammarco and Renzi2008) and Renzi (Reference Renzi2010) mostly considered only the first six modes, and Seo & Liu (Reference Seo and Liu2013) considered only the first 10 modes.

The analytical solution shines when the edge waves that propagate along $x=0$ or large-time/far-field solutions are of interest, where accurate numerical solutions are difficult to obtain. As mentioned previously, the numerical shoreline boundary treatment in linear theory is problematic. In addition, computational cost can be significant when the far-field or large-time solution is wanted, since numerical simulations have to resolve the entire domain starting from $t=0$ . Analytical solution is thus highly preferable in these situations. To examine the edge wave solutions more closely, in figure 14 we compare the analytical solutions with the numerical solution for the same double-Gaussian landslide forcing function case along $(x,y,t)=(0,y,5)$ . In this case, we find the $10$ -mode analytical solution to be nearly identical to the $15$ -mode solution, suggesting that consideration of the first $10$ modes is sufficient in this case (while not shown here, six modes are still insufficient in this example). We therefore advise that a convergence test be performed every time the analytical solution is used at a different location, to ensure that satisfactory convergence is reached within a reasonable number of modes considered. In figure 14, we also included $\unicode[STIX]{x1D702}_{extra}$ , and see that its amplitude is significant, again showing its permanent effect on the wave shape. On the other hand, even with the fine grid spacing $\unicode[STIX]{x0394}x=\unicode[STIX]{x0394}y=0.01$ , the numerical edge wave solutions still do not fully agree with the analytical solutions, which in this case have converged and should be regarded as the reference solution.

Figure 14.

Edge waves ( $x=0$ ) at $t=5$ due to a double-Gaussian landslide forcing function moving at a constant speed $1$ from $x=0$ : solid line, numerical solution with $\unicode[STIX]{x0394}x=\unicode[STIX]{x0394}y=0.01$ ; plus, $10$ -mode analytical solution; circle, $15$ -mode analytical solution; dashed line, $15$ -mode $\unicode[STIX]{x1D702}_{extra}$ .