Article Contents
Article Contents

# Nonlinear dispersion in wave-current interactions

• *Corresponding author: Ruiao Hu

DH is partially supported by ERC Synergy Grant 856408 - STUOD (Stochastic Transport in Upper Ocean Dynamics). RH is supported by an EPSRC scholarship [grant number EP/R513052/1]

• Via a sequence of approximations of the Lagrangian in Hamilton's principle for dispersive nonlinear gravity waves we derive a hierarchy of Hamiltonian models for describing wave-current interaction (WCI) in nonlinear dispersive wave dynamics on free surfaces. A subclass of these WCI Hamiltonians admits emergent singular solutions for certain initial conditions. These singular solutions are identified with a singular momentum map for left action of the diffeomorphisms on a semidirect-product Lie algebra. This semidirect-product Lie algebra comprises vector fields representing horizontal current velocity acting on scalar functions representing wave elevation. We use computational simulations to demonstrate the dynamical interactions of the emergent wavefront trains which are admitted by this special subclass of Hamiltonians for a variety of initial conditions.

In this paper, we investigate:

(1) A variety of localised initial current configurations in still water whose subsequent propagation generates surface-elevation dynamics on an initially flat surface; and

(2) The release of initially confined configurations of surface elevation in still water that generate dynamically interacting fronts of localised currents and wave trains. The results of these simulations show intricate wave-current interaction patterns whose structures are similar to those seen, for example, in Synthetic Aperture Radar (SAR) images taken from the space shuttle.

Mathematics Subject Classification: Primary: 76M30, 76M60; Secondary: 76N30.

 Citation:

• Figure 2.  Synthetic Aperture Radar (SAR) Image of internal-wave signatures on the South China Sea

Figure 3.  Custom colour map for the upcoming $|u|^2$ plots

Figure 4.  Locations of the 1D profiles of $|\boldsymbol{{u}}|$ and $\overline{D}$ in the 2D numerical simulations

Figure 5.  Evolution of $|u|^2$ with the "plate" initial condition with $\alpha = w_0$

Figure 6.  Evolution of $\overline{D}-\overline{b}$ with the "plate" initial condition with $\alpha = w_0$

Figure 7.  Evolution of $|u|^2$ with the "plate" initial condition with $\alpha = w_0/8$

Figure 8.  Evolution of $\overline{D}-\overline{b}$ with the "plate" initial condition with $\alpha = w_0/8$

Figure 9.  Evolution of $|u|^2$ with the "skew" initial condition with $\alpha = w_0$

Figure 10.  Evolution of $\overline{D}-\overline{b}$ with the "skew" initial condition with $\alpha = w_0$

Figure 11.  Evolution of $|u|^2$ with the "skew" initial condition with $\alpha = w_0/8$

Figure 12.  Evolution of $\overline{D}-\overline{b}$ with the "skew" initial condition with $\alpha = w_0/8$

Figure 13.  Evolution of $|u|^2$ with the "wedge" initial condition with $\alpha = w_0$

Figure 14.  Evolution of $\overline{D}-\overline{b}$ with the "wedge" initial condition with $\alpha = w_0$

Figure 15.  Evolution of |u|2 with the "wedge" initial condition with 8α = w0

Figure 16.  Evolution of $\overline{D}-\overline{b}$ with the "wedge" initial condition with 8α = w0

Figure 17.  Evolution of $|u|^2$ with the "parallel" initial condition with $\alpha = w_0$

Figure 18.  Evolution of $\overline{D}-\overline{b}$ with the "parallel" initial condition with $\alpha = w_0$

Figure 19.  Evolution of $|u|^2$ with the "parallel" initial condition with $8\alpha = w_0$

Figure 20.  Evolution of $\overline{D}-\overline{b}$ with the "parallel" initial condition with $8\alpha = w_0$

Figure 21.  Evolution of $\overline{D}-\overline{b}$ with the "dam break" initial condition with $\alpha = w_0$

Figure 22.  Evolution of $|u|^2$ with the "dam break" initial condition with $\alpha = w_0$

Figure 23.  Evolution of $\overline{D}-\overline{b}$ with the "dam break" initial condition with $8\alpha = w_0$

Figure 24.  Evolution of $|u|^2$ with the "dam break" initial condition with $8\alpha = w_0$

Figure 25.  Evolution of $\overline{D}-\overline{b}$ with the "dam break" initial condition with $\alpha = w_0$ and zero bathymetry

Figure 26.  Evolution of $|u|^2$ with the "dam break" initial condition with $\alpha = w_0$ and zero bathymetry

Figure 27.  Evolution of $\overline{D}-\overline{b}$ with the "dam break" initial condition with $8\alpha = w_0$

Figure 28.  Evolution of $|u|^2$ with the "dam break" initial condition with $8\alpha = w_0$

Figure 29.  Evolution of |u|2 with the "dual dam break" initial condition with α = w0

Figure 30.  Evolution of $\overline{D}-\overline{b}$ with the "dual dam break" initial condition with α = w0

Figure 31.  Evolution of |u|2 with the "dual dam break" initial condition with 8α = w0

Figure 32.  Evolution of $\overline{D}-\overline{b}$ with the "dual dam break" initial condition with 8α = w0

•  Animation of velocity for section 3.2.1 one_plate_usqr Animation of elevation for section 3.2.1 one_plate_elev Animation of velocity for section 3.2.2 skew_usqr Animation of elevation for section 3.2.2 skew_elev Animation of velocity for section 3.2.3 wedge_usqr Animation of elevation for section 3.2.3 wedge_elev Animation of velocity for section 3.2.4 parallel_usqr Animation of elevation for section 3.2.4 parallel_elev Animation of velocity for section 3.2.5 dam_break_usqr Animation of elevation for section 3.2.5 dam_break_elev Animation of velocity for section 3.2.5 with constant bathymetry dam_break_no_bathymetry_usqr Animation of elevation for section 3.2.5 with constant bathymetry dam_break_no_bathymetry_elev Animation of velocity for section 3.2.6 dual_dam_break_usqr Animation of elevation for section 3.2.6 dual_dam_break_elev Zip of all related files Nonlinear dispersion in wave-current interactions supplementary material
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