Viscous damping of gravity-capillary waves: Dispersion relations and nonlinear corrections

We discuss the impact of viscosity on nonlinear propagation of surface waves at the interface of air and a fluid of large depth. After a survey of the available approximations of the dispersion relation, we propose to modify the hydrodynamic boundary conditions to model both short and long waves. From them, we derive a nonlinear Schrödinger equation where both linear and nonlinear parts are modified by dissipation and show that the former plays the main role in both gravity and capillary-gravity waves while, in most situations, the latter represents only small corrections. This provides a justification of the conventional approaches to damped propagation found in the literature.

Figure 1

Dispersion relation for gravity-capillary waves propagating at the interface of air and water in the presence of kinematic viscosity. Numerical solution of Eq. (2) for forward-propagating waves (${\omega }_R\ge 0$). On the left axis (in logarithmic scale), the solid blue and dotted lines refer to $\mathrm{Re}\omega$ and its inviscid counterpart $\tilde{\omega }$. The red dotted and green dash-dotted lines pertain to the right axis (linear scale) and show the two branches of $\mathrm{Im}\omega$. The vertical dotted line marks to the cutoff $k_c$ beyond which ${\omega }_{\mathrm{R}}=0$ and ${\omega }_{\mathrm{I}}$ bifurcates in two standing-wave branches, yielding different damping rates.

Figure 2

Dispersion relation for gravity-capillary waves propagating at the interface of air and glycerin. (a) Real part; (b) imaginary part. Numerical solutions of Eq. (2) for forward-propagating waves (${\omega }_R\ge 0$) are shown as a solid dark yellow line. The conventional approximation is shown by a green dotted line. The different approximations are represented by dashed lines (black, DM; red, VPF; pink, MVPF). Finally, the dash-dotted lines correspond to the SDM: It does not exhibit a clear cutoff, but provides a good approximation of damping at both $k$ limits.

Figure 3

Comparison of the nonlinear evolution of MI with linear and nonlinear damping. Time is normalized by nonlinear time $T_0={\left(2{\varepsilon }^2{k}_0\right)}^{-1}$. $k_0=10, \varepsilon =0.1, \nu =1\times {10}^{-6}, \alpha =0.01$, and surface tension is neglected. (a) Oscillations of the central peak (Stokes wave); (b) spectral mean ${\kappa }_{\mathrm{m}}$. The solid blue lines represent the full model [Eq. (17)], and the dashed red (resp., black dotted) lines are obtained by neglecting $\widehat{D}\left(k\right)$ in the denominators of the nonlinear (resp., linear) part.

Figure 4

Comparison of results with purely linear (solid lines) and purely nonlinear (dashed lines) damping. Parameters are $k_0=75, \varepsilon =0.1, \nu =1\times {10}^{-6}, s=7.28\times {10}^{-5}$, and $\alpha =0.01$. (a) Central mode $\kappa =0$; (b) main unstable modes $\kappa =\pm {\kappa }_0$. We notice that the downshifted mode at $-{\kappa }_0$ soon acquires most of the energy. While the linear damping irreversibly stops recurrence, the nonlinear damping is not sufficient.

Figure 5

Evolution in time of $SN-PQ$ (blue solid line) and $2k_0(QN-P^2)+\left(SN-PQ\right)$ (red dashed line) for surface waves at the air-water interface. Panel (a) [resp., (b)] corresponds to the gravity (resp., gravity-capillary) case of Fig. 3 [resp., Fig. 4] in the main text. While $SN-PQ$ is not always rigorously positive, so it is in most of the situations and the net mean-frequency shift, as predicted by Eq. (A4) is always positive.