We formulate the twodimensional gravitycapillary water wave equations in a spatially quasiperiodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudospectral discretization of the equations of motion in which onedimensional quasiperiodic functions are represented by twodimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasiperiodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of timestepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential timedifferencing (ETD) scheme. The ETD approach makes use of the smallscale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasiperiodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasiperiodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.
 Award ID(s):
 1716560
 NSFPAR ID:
 10312807
 Date Published:
 Journal Name:
 Journal of Fluid Mechanics
 Volume:
 915
 ISSN:
 00221120
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract 
We present a numerical study of spatially quasiperiodic gravitycapillary waves of finite depth in both the initial value problem and travelling wave settings. We adopt a quasiperiodic conformal mapping formulation of the Euler equations, where onedimensional quasiperiodic functions are represented by periodic functions on a higherdimensional torus. We compute the time evolution of free surface waves in the presence of a background flow and a quasiperiodic bottom boundary and observe the formation of quasiperiodic patterns on the free surface. Two types of quasiperiodic travelling waves are computed: smallamplitude waves bifurcating from the zeroamplitude solution and largeramplitude waves bifurcating from finiteamplitude periodic travelling waves. We derive weakly nonlinear approximations of the first type and investigate the associated smalldivisor problem. We find that waves of the second type exhibit striking nonlinear behaviour, e.g. the peaks and troughs are shifted nonperiodically from the corresponding periodic waves due to the activation of quasiperiodic modes.

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