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Wave-Current-Bathymetry Interaction Revisited: Modeling, Analysis and Asymptotics

Adrian Kirkeby, Trygve Halsne

Abstract

Surface gravity waves propagating over variable currents and bathymetry are studied in the linear regime. The leading-order problem is formulated in terms of surface variables, surface current, and the bathymetry-dependent Dirichlet-to-Neumann (DN) operator. Well-posedness of the governing PDE is then established using the theory of hyperbolic systems of pseudo-differential operators. The asymptotic analysis of waves in a slowly varying environment is subsequently considered: the semiclassical Weyl quantization of the symbol $g_b(X,ξ)=|ξ|\tanh(b(X)|ξ|)$ is shown to be both asymptotically accurate and consistent with the self-adjoint structure of the DN operator, and key properties needed for the asymptotic analysis are derived. The energy dynamics are then examined, leading to a novel equation for the evolution of total wave energy. Moreover, the asymptotic surface model based on the Weyl quantization of the DN operator is shown to recover well-known asymptotic models such as the wave action equation, the mild-slope equation, the Schrödinger equation, and the action balance equation, thereby providing a unified framework for the linear theory of wave-current-bathymetry interaction. Several numerical experiments are included throughout to illustrate the theoretical results.

Wave-Current-Bathymetry Interaction Revisited: Modeling, Analysis and Asymptotics

Abstract

Surface gravity waves propagating over variable currents and bathymetry are studied in the linear regime. The leading-order problem is formulated in terms of surface variables, surface current, and the bathymetry-dependent Dirichlet-to-Neumann (DN) operator. Well-posedness of the governing PDE is then established using the theory of hyperbolic systems of pseudo-differential operators. The asymptotic analysis of waves in a slowly varying environment is subsequently considered: the semiclassical Weyl quantization of the symbol is shown to be both asymptotically accurate and consistent with the self-adjoint structure of the DN operator, and key properties needed for the asymptotic analysis are derived. The energy dynamics are then examined, leading to a novel equation for the evolution of total wave energy. Moreover, the asymptotic surface model based on the Weyl quantization of the DN operator is shown to recover well-known asymptotic models such as the wave action equation, the mild-slope equation, the Schrödinger equation, and the action balance equation, thereby providing a unified framework for the linear theory of wave-current-bathymetry interaction. Several numerical experiments are included throughout to illustrate the theoretical results.

Paper Structure

This paper contains 14 sections, 12 theorems, 217 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Well-Posedness of the Wave-Current-Bathymetry System
  4. Pseudodifferential Analysis
  5. Analysis of the Wave System
  6. Asymptotic Analysis: Energy, Wave Action and Diffraction
  7. Asymptotics of the DN-operator
  8. Asymptotic Energy Dynamics
  9. Numerical Examples
  10. Diffractive models
  11. Numerical examples
  12. Beyond WKB: phase-space dynamics
  13. Numerical examples
  14. Conclusion

Key Result

Proposition 1

Assume that $\nabla_{X,z} \times \bar{\bm{U}} = \bar{\bm{\omega}} = \mathcal{O}(\delta)$ and that there is some fixed $R \geq 0$ such that $\bar{\bm{\omega}} = 0$ for $|X| \geq R$. Moreover, assume the wave perturbation satisfies $D^\alpha\bm{u} = \mathcal{O}(\varepsilon)$ for $|\alpha| \leq 3$ and

Figures (5)

  • Figure 1: The figure illustrate the main components of our model; propagating waves (top), current (middle) and bathymetry (lower). Although the scales have been compressed for the purpose of illustration, the depicted current and bathymetry are used in the numerical simulation of the plotted wave in the numerical investigation of diffractive effects in \ref{['sect: diffraction']}.
  • Figure 2: The first panel shows a snapshot of the propagating wave $\eta$ on top of the bulk current. In the bulk the $z$-direction is compressed for economic reasons, and the vertical velocity is slightly amplified for visibility. The second panel shows evolution of $E_T(t)$ and $\tilde{E}_T(t)$ throughout the simulation, while the third panel shows the computed values of $I_s(t)$ and $I_b(t)$.
  • Figure 3: In the first panel we see the amplitude $A_\eta(t,X) = \sqrt{E/g}$ and the simulated wave packet $\eta(t,X)$ together with the bathymetry and 1D current. The vertical dashed lines indicate the measurement domain $\mathcal{D}_m$. In the second panel we plot the maxima and relative difference of $E$ and $\mathcal{E}$ in $\mathcal{D}_m$ as a function of time. The vertical dashed lines indicates approximately when the peak of the wave packet enters and leaves $\mathcal{D}_m$. The third panel shows the time evolution of the total energy of both $E$ and $\mathcal{E}$, both normalized by $\|E_0\|_{L^1}$
  • Figure 4: The left panels show the pointwise difference $\mathcal{E}(t,X) -E(t,X)$ at three different times. The lack of diffraction in $E$ manifests itself as overestimation of $\mathcal{E}$ around the peak, while underestimating it away from the peak. The pointwise difference $\mathcal{E}(t,X) -E_S(t,X)$ for the Schrödinger energy density shown in the right panels does not have a clear interpretation, but is about a factor $10^{-1}$ smaller. The lower panel shows the maximum evolution of approximation error as the wave propagates through $\mathcal{D}_m$ for the two densities.
  • Figure 5: The top panel shows several snapshots of the evolution of $\mathcal{E}$. Initially, the wave propagates to the right, before it stops and eventually starts propagating backwards at $x_1 \approx 1240 \ m$. The energy density $\mathcal{E}$ grows rapidly around the stopping point, and continues to grow as the wave starts propagating to the left. In the lower panel, the behavior is seen in phase space; the trajectory $(x(t),k(t))$ changes direction at the stopping point, and the wavenumber keeps increasing as the wave propagates against the current. The color represents the value of $E$ along the trajectory, and matches the simulated $\mathcal{E}$ quite well. The middle panels shows the Wigner distribution at three different times; one can clearly see how it is concentrated around the phase space coordinates $(x(t),k(t))$ from the characteristics, and this becomes even more apparent in the lower panel, where we have plotted the coordinates of $\text{argmax}_{x,k} W(\psi)$.

Theorems & Definitions (23)

  • Proposition 1
  • Proposition 2
  • Definition 1
  • Lemma 1
  • Proposition 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 13 more