Admissibility of Solitary Wave Modes in Long-Runout Debris Flows

TL;DR

The paper develops a depth-averaged, non-Newtonian debris-flow model with curvature-induced dispersion and rheology-dependent coefficients, deriving a Korteweg–de Vries (KdV) reduction that admits both cnoidal waves and solitary waves. It maps regimes in a Froude–slope diagram, showing dispersive pulses are admissible on gentle slopes while roll-waves dominate steeper terrains, and validates the reduced model against full rheological simulations where cnoidal and soliton waves persist with speeds and widths predicted by the KdV theory. A practical nonlinearity diagnostic is introduced, allowing estimation of the effective nonlinearity from crest speeds and thicknesses; velocity-based proxies are provided when crest speeds are unavailable, with caveats near zero. The results indicate dispersive pulses can complement roll-wave dynamics by contributing to momentum transport on gentle reaches, but do not universally explain long runout; the framework also highlights regimes where higher-order corrections and multi-dimensional effects may be important for real debris-flow events.

Abstract

Debris flows often exhibit coherent wave structures-shock-like roll waves on steeper slopes and weaker, more sinusoidal dispersive pulses on gentler slopes. Coarse-rich heads raise basal resistance, whereas fines-rich tails lower it; in gentle reaches, small-amplitude pulses can locally transport momentum across low-resistance segments. We focus on this gentle-slope, long-wave, low-amplitude regime, where the base-flow Froude number is order unity. In this limit, we obtain a Korteweg-de Vries (KdV) reduction from depth-averaged balances with frictional (Coulomb) and viscous-plastic basal options, using a curvature-type internal normal-stress closure in the long-wave small-k regime. Multiple-scale analysis yields effective nonlinear and dispersive coefficients. We also introduce a practical nonlinearity diagnostic that can be computed from observed crest speeds and flow thicknesses. When laboratory-frame crest celerity is available, we estimate an effective quadratic coefficient from the KdV speed-amplitude relation and report its ratio to the shallow-water reference. When only a depth-averaged first-surge speed and thickness are available, we use the same construction to form a velocity-based proxy and note its bias near zero. A Froude-slope diagram organizes published cases into a steep-slope roll-wave domain and a gentle-slope corridor where KdV pulses are admissible. Numerical solutions of the full depth-averaged model produce cnoidal and solitary waves that agree with the reduced KdV predictions within this corridor. We regard dispersive pulses as a regime-specific complement to roll-wave dynamics, offering a condition-dependent contribution to mobility on gentle reaches rather than a universal explanation for long runout.

Figures (9)

• Figure 1: Relationships among (a) apparent coefficient of friction ($H_{\mathrm{max}}/L_{\mathrm{max}}$), (b) runout distance ($L_{\mathrm{max}}$), (c) deposit area ($A$), and event volume ($V$) for 203 landslides and debris flows compiled by Legros Legros2002. Only debris-flow and subaerial cases are used in the quantitative analysis of Sec. \ref{['sec:soliton_energy']}; submarine and planetary cases are shown for completeness. Symbols distinguish subaerial non-volcanic landslides (nv), subaerial volcanic landslides (v), submarine landslides, Martian landslides, and debris flows. Some entries lack estimates for one or more parameters. Each panel uses the subset of points with available data for the plotted quantities. The dataset is from Legros Legros2002, supplemented with values from Iverson Iverson1997.
• Figure 2: (a) Iso-work contours of the distal work magnitude $|W|$; (b) the same budget partitioned by the tail-averaged basal Coulomb coefficient $\mu_{\mathrm{tail}}$. Both panels use Eq. (\ref{['eq:work']}) with $W=\rho g h_{\mathrm{tail}} W_b L^{2}\,|S-\mu_{\mathrm{tail}}|$ and the parameter grid described in the text. Take-home: $|W|$ scales linearly with $h_{\mathrm{tail}}$ and quadratically with $L$; near $S\approx\mu_{\mathrm{tail}}$, small changes in $\mu_{\mathrm{tail}}$ cause large relative changes in $|W|$.
• Figure 3: Froude–slope $(\mathrm{Fr},S)$ regime diagram for the compiled cases. The solid curve $S_{\mathrm{rw}}=0.07/\mathrm{Fr}^{2}$ is a guide to the eye for this dataset—an empirical organizer, not a universal threshold nor a direct consequence of linear Saint–Venant roll-wave theory. Boundaries can shift with rheology, bed roughness, channel geometry, and boundary conditions. The horizontal dashed line at $S=0.03$ marks the lower bound of the “dispersive-pulse corridor” used here to contextualize gentle reaches; it is likewise dataset-specific. Symbols: solid circles = head-reach field points; open circles = distal field points; solid triangles = laboratory flume runs (see legend). Numerical ranges and sources for all symbols are listed in Table \ref{['tab:merged_fr_s']}. Here $\mathrm{Fr}$ is the event-scale Froude number computed from published $u$–$h$ pairs; $\mathrm{Fr}_0$ is reserved for the base-flow value used in the asymptotics and in the nonlinearity proxy $\tilde{\alpha}_{\mathrm{norm}}$ (Figure \ref{['fig:alphaFr']}). Interpretation note: this panel provides regime context; “KdV-admissible” in the text additionally requires the dispersion-polarity choice $\beta_{\mathrm{eff}}>0$ and a positive estimated nonlinearity $\alpha_{\mathrm{eff}}^{\mathrm{lab}}>0$ (or proxy sign when $V$ is unavailable). Heterogeneous measurement methods preclude uniform error bars, so points should be viewed as representative rather than precise thresholds.
• Figure 4: Semilog plot of $\log_{10} |\tilde{\alpha}_{\mathrm{norm}}|$ versus base-flow Froude number $\mathrm{Fr}_0=u_0/\sqrt{g h_0}$. The plotted quantity is a velocity-based proxy with $\tilde{\alpha}_{\mathrm{eff}}^{\mathrm{lab}}=3(u_{1s}-u_0-c_0)/A$, $\alpha_0^{\mathrm{lab}}=3c_0/(2h_0)$, and $\tilde{\alpha}_{\mathrm{norm}}=\tilde{\alpha}_{\mathrm{eff}}^{\mathrm{lab}}/\alpha_0^{\mathrm{lab}}$, where $A=h_{1s}-h_0$ and $c_0$ as defined in § \ref{['sec:notation']}. Markers show $|\tilde{\alpha}_{\mathrm{norm}}|$; filled markers indicate $\tilde{\alpha}_{\mathrm{eff}}^{\mathrm{lab}}<0$. Straight-line guides summarize apparent trends: for $\mathrm{Fr}_0<1$, $\log_{10}|\tilde{\alpha}_{\mathrm{norm}}| \approx -2.9 \mathrm{Fr}_0$; for $\mathrm{Fr}_0>1$, $\log_{10}|\tilde{\alpha}_{\mathrm{norm}}| \approx 0.3 \mathrm{Fr}_0$. These lines are derived from semi-log linear regressions between $\log_{10} |\tilde{\alpha}_{\mathrm{norm}}|$ and $\mathrm{Fr}_0$. Negative proxy values commonly occur when $u_{1s}<u_0+c_0$ for $A>0$; because the proxy substitutes $u_{1s}$ for the crest speed $V$, it can underestimate $\alpha_{\mathrm{eff}}^{\mathrm{lab}}$, so proxy sign alone is not used as a KdV-admissibility criterion (celerity-based estimates are preferred where available). One sample from Rio Moscardo (20 Jul 1993) is not included in the analysis due to a very small magnitude close to zero (e.g., $<10^{-3}$).
• Figure 5: Cnoidal wave propagation under the full rheological model (left) compared to the ideal cnoidal wave solution (right). The reference waveform is constructed using the same effective coefficients and wavelength as in the simulation, enabling a like-for-like comparison. The sustained periodicity and stable waveform confirm that under weakly nonlinear, long-wavelength conditions, cnoidal waves closely match their KdV counterparts.