The Kadomtsev-Petviashvili equation in conformal variables for waves over topography

This paper extends the conformal mapping approach to weakly transversal surface waves over topography by deriving a Kadomtsev-Petviashvili type equation in conformal variables that accommodates non-smooth bottom features through the use of effective depth, thereby unifying and extending existing weakly nonlinear dispersive wave models.

The Gist

Imagine you are watching a wave travel across a lake. Usually, we think of the lake floor as a smooth, gentle slope. But in the real world, the bottom is often rough: it has jagged rocks, sudden drop-offs, or even a "staircase" of submerged ridges.

For a long time, scientists had a hard time mathematically describing how waves move over these rough bottoms. The standard math tools required the bottom to be a perfectly smooth curve, like a sliding hill. If the bottom was jagged or "broken," the math would break down.

This paper introduces a clever new way to solve that problem, allowing us to model waves over any kind of bottom, no matter how rough or weird it looks.

Here is the breakdown of their discovery using simple analogies:

1. The Problem: The "Smoothie" vs. The "Chunky" Bottom

Think of the ocean floor as a piece of terrain.

• The Old Way: To predict how a wave moves, scientists used a map that only worked if the terrain was a smooth, continuous line. If you tried to map a jagged cliff or a step-like trench, the math got stuck. It was like trying to roll a ball over a staircase; the ball (the math) just wouldn't work.
• The New Way: The authors use a technique called Conformal Mapping. Imagine you have a crumpled piece of paper (the rough ocean floor). Instead of trying to measure the crumples directly, you iron the paper flat.
  • In this "ironed" world, the jagged rocks and trenches are smoothed out into a gentle, uniform landscape.
  • The wave travels easily over this smooth, ironed landscape.
  • Once the calculation is done, they "un-iron" the paper to see what the wave looks like back in the real, rough world.

2. The Secret Ingredient: The "Effective Depth"

The magic of this method is a concept the authors call "Effective Depth."

In the real world, the depth might change instantly from 10 meters to 2 meters (a cliff). In the "ironed" world, that cliff becomes a gentle, slow slope.

• The authors realized that the wave doesn't actually "feel" the jagged edges of the cliff. Instead, it feels the average or smoothed-out version of the depth.
• They call this the Effective Depth. It's like how a car driving over a bumpy road feels the average height of the bumps, not every single pebble.
• The Big Discovery: You can use this "Effective Depth" in the standard wave equations, even if the real bottom is made of sharp rectangles, jagged rocks, or anything that isn't a smooth function. The math works because the wave "sees" the smooth version.

3. The New Equations: The "Wave Traffic Report"

The authors derived two new mathematical formulas (called KP equations) to predict how these waves behave.

• Equation A (The Slow Change): This is for when the water depth changes gradually, like a long, gentle slope. It's like a traffic report for a highway that slowly gets narrower.
• Equation B (The Small Bumps): This is for when the bottom has small, gentle ripples. It's like a traffic report for a road with small speed bumps.

Both equations tell us how a wave will speed up, slow down, or break apart as it hits different parts of the ocean floor.

4. The Simulation: The "Digital Ocean"

To prove their math works, the scientists ran computer simulations.

• The Setup: They created a digital ocean with a bottom made of a series of sharp, rectangular blocks (like a staircase underwater). This is the "rough" bottom that usually breaks math models.
• The Test: They sent a wave packet (a group of waves) through this digital ocean.
• The Result:
  • The wave slowed down as it hit the "stairs."
  • It created a "wake" (a trailing ripple) behind it.
  • Most importantly, their new equations predicted exactly how the wave would behave, whereas older models would have struggled or failed with such a jagged bottom.

Why Does This Matter?

This is a big deal for a few reasons:

1. Realism: Real ocean floors are rarely smooth. They have shipwrecks, reefs, and underwater mountains. This new math can handle that reality without needing to pretend the ocean floor is smooth.
2. Safety: Better wave models mean better predictions for tsunamis and storm surges hitting coastlines with complex underwater geography.
3. Simplicity: It allows scientists to use simpler, faster computer models to solve problems that previously required incredibly complex and slow simulations.

In a nutshell: The authors found a way to "iron out" the rough ocean floor in their math, allowing them to predict how waves travel over jagged, rocky, or weirdly shaped bottoms with high accuracy. They proved that the wave cares more about the smooth average of the depth than the jagged details of the rocks.

Technical Summary

1. Problem Statement

The paper addresses the challenge of modeling weakly nonlinear, weakly dispersive, and weakly transversal surface waves propagating over complex, non-smooth topography (bathymetry).

• Limitation of Existing Methods: Traditional conformal mapping techniques are powerful for 2D potential flows but are naturally restricted to single horizontal spatial variables. Extending these to 3D free-surface flows is computationally expensive.
• Limitation of Reduced Models: Standard reduced models (like the KdV or KP equations) typically require the topography to be a smooth function or satisfy specific "slowly varying" or "small amplitude" hypotheses in the physical domain. They struggle to handle "rough" topographies (e.g., step-like changes, trenches, or discontinuous functions) without losing accuracy or requiring complex smoothing.
• Goal: To derive a Kadomtsev–Petviashvili (KP) type equation formulated in conformal variables that can accurately model wave propagation over arbitrary topographies, including those that are not smooth or even functions in the classical sense.

2. Methodology

The authors employ a multi-step mathematical framework combining conformal mapping, potential theory, and asymptotic analysis:

A. Conformal Mapping Extension

• Framework: Building on previous work (Andrade & Nachbin, 2018), the authors extend the conformal mapping technique to a 3D domain.
• Transformation: They map the physical fluid domain (bounded by a free surface $z=\eta$ and a ridge-like bottom $z=-\mu(1+H(x))$) to a uniform strip in the complex plane.
• Key Variable: The mapping introduces a Jacobian, denoted as $M(\xi)$ (or $J$), which encodes the topographic information. Crucially, $M(\xi)$ is a smooth function derived from the convolution of the topography with a kernel, even if the physical topography $H(x)$ is discontinuous.

B. Governing Equations in Harmonic Coordinates

• The full 3D Euler equations for an ideal, incompressible, irrotational flow are rewritten in the new harmonic coordinates $(\xi, y, \zeta)$.
• This results in a system where the bathymetry effects are incorporated into variable coefficients ($M(\xi)$, $I(\xi)$) and a modified differential operator $\tilde{\Delta}$.
• A Terrain-Following Boussinesq System is derived in these coordinates to serve as the intermediate model before asymptotic expansion.

C. Asymptotic Derivation of KP Equations

The authors perform asymptotic expansions under the standard KP scaling ($\varepsilon = O(\mu^2)$, $\gamma = O(\mu)$) to derive two distinct KP equations based on different assumptions about the topography:

1. Case 1: Slowly Varying Depth (General Case)

   • Assumption: The Jacobian $M(\xi)$ is a slowly varying function of the slow variable $\mu^2 \xi$.
   • Derivation: Using Riemann invariants and perturbation expansions, they derive a KP equation with variable coefficients dependent on $M(\xi)$.
   • Resulting Equation: A generalized KP equation (Eq. 41) where the wave speed and dispersion terms depend on the "effective depth" $d(x) = M(\xi(x))$.

2. Case 2: Small Amplitude Topography

   • Assumption: The topography has small amplitude, allowing $M(\xi)$ to be expanded as $1 + \varepsilon m(\xi)$.
   • Derivation: Utilizing a recent expansion technique (Ludu et al., 2025), they derive a simplified KP equation (Eq. 53) suitable for low-correlation topographies.

3. Key Contributions

• Handling "Rough" Topography: The most significant contribution is the decoupling of the smoothness requirement from the physical topography. The model allows for discontinuous or non-smooth physical bathymetries (like rectangular trenches shown in Figure 1) because the governing equations rely on the effective depth $d(x)$, which is the smooth Jacobian of the conformal map.
• Two New KP Models:
  1. A variable-coefficient KP equation for slowly varying depths, which generalizes existing models (like those by Grimshaw and Johnson) by replacing physical depth with effective depth.
  2. A small-amplitude KP equation that extends the work of Ludu et al. to include transversal effects.
• Consistency: The derived equations recover the classical KP and KdV equations when the depth is constant ($M=1$), ensuring mathematical consistency.
• Physical Interpretation: The paper clarifies that "effective depth" is the true parameter governing wave dynamics in reduced models, explaining why reduced models often work well even with non-smooth topographies if the correct effective depth is used.

4. Results

• Numerical Simulations: The authors performed numerical simulations using pseudo-spectral methods (FFT) to solve the derived equations.
• Test Case: A localized Gaussian pulse was propagated over a patch of rectangular topography (a "staircase" profile).
• Comparative Analysis:
  • Equation (41) (Slowly Varying): Captured the wave slowing down and the generation of an oscillatory trailing wake due to the interaction between the topography and dispersion.
  • Equation (53) (Small Amplitude): Showed a different wake structure compared to Eq. (41), highlighting the sensitivity of the model to the specific topographic assumptions.
  • Classical KP: Served as a baseline, showing that without topographic terms, the wave propagates without the specific slowing or wake modifications observed in the other models.
• Observation: The simulations confirmed that the variable topography significantly alters wave speed and shape, and the choice of the specific KP formulation (slowly varying vs. small amplitude) affects the details of the wake generation.

5. Significance and Implications

• Theoretical Advancement: This work bridges the gap between rigorous conformal mapping techniques (usually 2D) and 3D reduced wave models. It provides a mathematically consistent way to include complex bathymetry in KP/KdV models without ad-hoc smoothing of the physical domain.
• Practical Application: The findings suggest that hydrodynamic models for tsunamis, coastal waves, or internal waves can utilize "rough" bathymetric data (e.g., from sonar scans with steps or discontinuities) directly by converting them into an effective depth profile. This removes the need for artificial smoothing of the seabed, which can introduce errors in wave speed and energy conservation.
• Future Utility: The framework opens the door for more accurate and efficient numerical simulations of wave-topography interactions in complex environments, such as continental shelves with terraces or submerged ridges, using computationally cheaper reduced models rather than full 3D Euler solvers.

In summary, Andrade and Flamarion successfully extended the conformal mapping approach to derive robust KP equations that accommodate arbitrary topographies by utilizing the concept of effective depth, thereby enhancing the applicability of reduced wave models to real-world, complex oceanic environments.