Dispersive waves generated by an underwater landslide

In this work we study the generation of water waves by an underwater sliding mass. The wave dynamics are assumed to fell into the shallow water regime. However, the characteristic wavelength of the free surface motion is generally smaller than in geophysically generated tsunamis. Thus, dispersive effects need to be taken into account. In the present study the fluid layer is modeled by the Peregrine system modified appropriately and written in conservative variables. The landslide is assumed to be a quasi-deformable body of mass whose trajectory is completely determined by its barycenter motion. A differential equation modeling the landslide motion along a curvilinear bottom is obtained by projecting all the forces acting on the submerged body onto a local moving coordinate system. One of the main novelties of our approach consists in taking into account curvature effects of the sea bed.

💡 Research Summary

In this paper the authors investigate the generation of surface waves by an underwater landslide (submarine mass slide) in a shallow‑water setting where the characteristic wavelength is short enough that dispersive effects become important. To capture both the wave dynamics and the motion of the sliding mass, they couple a modified Peregrine (Boussinesq‑type) system written in conservative variables with a dynamical model for the landslide that explicitly accounts for the curvature of the seabed.

The fluid model starts from the classic Peregrine equations, but the authors extend them to allow a time‑dependent bottom elevation d(x,t). Using the total water depth H = d + η and the depth‑averaged discharge Q = H u as conserved variables, the mass equation retains its simple form H_t + Q_x = 0, while the momentum equation acquires a source term ½ H d̈x that represents the vertical acceleration of the moving bottom. The linear dispersion relation of the modified system is identical to that of the original Peregrine model; only the nonlinear terms differ, which means the model remains accurate for weakly nonlinear, weakly dispersive waves while incorporating the effect of a moving seabed.

For the landslide, the authors assume a quasi‑deformable solid of known shape and constant volume. The slide’s motion is reduced to the trajectory of its barycentre x_c(t). By introducing an arc‑length coordinate s along the seabed and a local tangent‑normal basis (τ, n), they project all forces onto the tangent direction. The total horizontal force consists of (i) the net gravity‑buoyancy component, (ii) a quadratic water‑drag term, (iii) a Coulomb‑type friction proportional to the normal force, and (iv) a curvature‑induced centrifugal term proportional to κ(x)(ds/dt)², where κ(x) is the signed curvature of the static bathymetry. The resulting second‑order ordinary differential equation (Eq. 3.3) reads

 (γ + c_w) S d²s/dt² = (γ − 1) g I₁(t) − c_f σ(t) I₂(t) − σ(t) c_f γ I₃(t) + ½ c_d A σ(t)(ds/dt)²,

with γ = ρ_ℓ/ρ_w > 1, and the integrals I₁, I₂, I₃ representing the projection of the slide shape onto sin θ, cos θ and curvature, respectively. Added mass is included via the coefficient c_w. Initial conditions are s(0)=0 and ṡ(0)=0.

Numerically, both the fluid and slide equations are integrated with a third‑order Bogacki‑Shampine Runge–Kutta scheme. The fluid equations are discretized by a finite‑volume method with UNO2 reconstruction for the advective terms and central differences for the dispersive terms; each time step requires solving a tridiagonal system for Q_t. The slide integrals are evaluated with the trapezoidal rule.

The test case uses a one‑dimensional domain x∈