Finite-amplitude wave propagation in a stratified fluid of variable depth

Variable-coefficient Korteweg - de Vries equation is applied to describe the interfacial wave transformation in two-layer fluid of variable depth. The soliton dynamics in this fluid is studied. The solitary wave breaks in two transient points. One of them is the point when two-layer fluid transforms to the on-layer flow. The second one is the point where layer thickness are equaled. The soliton amplitude dependence on the thickness of lower layer is found.

💡 Research Summary

The paper presents a comprehensive theoretical framework for the transformation of finite‑amplitude internal waves in a stratified two‑layer fluid whose depth varies slowly in the horizontal direction. Starting from the full Euler equations under the Boussinesq approximation, the authors derive a generalized Korteweg‑de Vries (KdV) model that incorporates both quadratic and cubic nonlinearities as well as dispersion. The coefficients of the model are obtained from the vertical density profile ρ₀(y) by solving a Sturm‑Liouville eigenvalue problem for the modal function Φ(y) and a nonlinear correction T(y). In the special case of a two‑layer fluid with constant densities ρ₁, ρ₂, explicit formulas for the linear wave speed c, the quadratic nonlinearity α, the cubic nonlinearity α₁, and the dispersion coefficient β are given (equations 15‑19).

When the bathymetry varies slowly, the authors extend the model to a Gardner equation with spatially varying coefficients (equations 9‑12). This equation conserves both mass and energy (equations 13‑14), which allows the use of the energy integral to track the evolution of solitary‑wave (soliton) parameters as the wave propagates over changing depth. The solitary‑wave solution of the Gardner equation is written in a compact form (equations 20‑21) involving a width parameter γ and a non‑dimensional parameter B that measures the relative strength of the quadratic and cubic nonlinearities. For the two‑layer case α₁<0, so B lies between 0 and 1; B→1 recovers the classic KdV soliton, while B→0 yields a “thick” soliton of infinite width.

A central result of the study is an analytical expression for the soliton amplitude a(x) as a function of the local lower‑layer thickness h₂(x), the initial lower‑layer thickness h₂₀, and the ratio q = h₁/h₂₀ (equation 29). This relation shows that the amplitude can either monotonically decay or exhibit a non‑monotonic behavior depending on q. The depth at which the amplitude reaches its maximum, h*, is given by equation (30), and the corresponding maximal amplification factor a*/a₀ by equation (31). If q > 0.5 (the upper layer is relatively thin), the amplitude continuously decreases as the wave approaches the critical point where the two layers become equal in thickness. If q < 0.5 (the upper layer is relatively thick), the amplitude first grows, reaches a peak, and then declines to zero.

Two distinct critical points are identified where the soliton “breaks”. The first occurs when the two layers have equal thickness (h₁ = h₂). At this point the quadratic nonlinearity coefficient α₁ passes through zero, causing the original soliton (of a given polarity) to vanish and a new soliton of opposite polarity to be generated—a phenomenon previously reported in the literature (Knickerbocker & Newell, 1980; Talipova et al., 1997). The second critical point is reached when the lower layer disappears (h₂ → 0), i.e., the two‑layer flow collapses into a single‑layer flow. Internal waves cannot exist in a single‑layer fluid, so the soliton is completely destroyed. Near this second point the soliton width diverges and the asymptotic analysis breaks down; nevertheless, the amplitude formally diverges as a ∝ h₂^{-2/3} (equation 32), indicating a potential “wave‑runup” or overturning zone in realistic oceanic settings.

Numerical illustrations (Figures 2‑4) demonstrate how the amplification factor depends on q, and how the soliton amplitude varies with the decreasing lower‑layer thickness for different initial thickness ratios (h₁/h₂₀ = 0.1, 0.5, 2, 6). The plots confirm the analytical predictions: for small q the amplitude first increases dramatically before vanishing, whereas for larger q the amplitude decays monotonically.

The authors emphasize that the whole procedure can be automated using a computational package they previously developed (Tugin et al., 2011), which evaluates the Sturm‑Liouville problem and the resulting coefficients directly from observed hydrographic data (density and bathymetry). Consequently, the model is ready for practical applications such as forecasting internal‑wave transformation over continental shelves, assessing the risk of wave breaking near steep slopes, and estimating energy transfer from internal tides to mixing.

In conclusion, the paper successfully extends the Gardner‑type solitary‑wave theory to realistic, horizontally varying stratified environments, identifies the precise conditions under which a solitary internal wave will either amplify, decay, or completely break, and provides explicit formulas for the amplitude evolution in terms of measurable physical parameters. Future work is suggested to incorporate higher‑order nonlinearities, mode coupling, and viscous dissipation to capture the dynamics in the immediate vicinity of the critical points more accurately.